/*
 * Copyright (C) 2007 The Guava Authors
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 * http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package com.google.common.collect;

import static com.google.common.base.Preconditions.checkArgument;
import static com.google.common.base.Preconditions.checkNotNull;
import static com.google.common.collect.CollectPreconditions.checkNonnegative;

import com.google.common.annotations.Beta;
import com.google.common.annotations.GwtCompatible;
import com.google.common.annotations.GwtIncompatible;
import com.google.common.base.Predicate;
import com.google.common.base.Predicates;
import com.google.common.collect.Collections2.FilteredCollection;
import com.google.common.math.IntMath;
import com.google.errorprone.annotations.CanIgnoreReturnValue;
import java.io.Serializable;
import java.util.AbstractSet;
import java.util.Arrays;
import java.util.BitSet;
import java.util.Collection;
import java.util.Collections;
import java.util.Comparator;
import java.util.EnumSet;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedHashSet;
import java.util.List;
import java.util.Map;
import java.util.NavigableSet;
import java.util.NoSuchElementException;
import java.util.Set;
import java.util.SortedSet;
import java.util.TreeSet;
import java.util.concurrent.ConcurrentHashMap;
import java.util.concurrent.CopyOnWriteArraySet;
import java.util.function.Consumer;
import java.util.stream.Collector;
import java.util.stream.Stream;
import org.checkerframework.checker.nullness.qual.MonotonicNonNull;
import org.checkerframework.checker.nullness.qual.Nullable;

Static utility methods pertaining to Set instances. Also see this class's counterparts Lists, Maps and Queues. 
See the Guava User Guide article on  Sets.
Author:
Kevin Bourrillion, Jared Levy, Chris Povirk
Since:
2.0/**
 * Static utility methods pertaining to {@link Set} instances. Also see this class's counterparts
 * {@link Lists}, {@link Maps} and {@link Queues}.
 *
 * <p>See the Guava User Guide article on <a href=
 * "https://github.com/google/guava/wiki/CollectionUtilitiesExplained#sets"> {@code Sets}</a>.
 *
 * @author Kevin Bourrillion
 * @author Jared Levy
 * @author Chris Povirk
 * @since 2.0
 */
@GwtCompatible(emulated = true)
public final class Sets {
  private Sets() {}

  
AbstractSet substitute without the potentially-quadratic removeAll implementation. /**
   * {@link AbstractSet} substitute without the potentially-quadratic {@code removeAll}
   * implementation.
   */
  abstract static class ImprovedAbstractSet<E> extends AbstractSet<E> {
    @Override
    public boolean removeAll(Collection<?> c) {
      return removeAllImpl(this, c);
    }

    @Override
    public boolean retainAll(Collection<?> c) {
      return super.retainAll(checkNotNull(c)); // GWT compatibility
    }
  }

  
Returns an immutable set instance containing the given enum elements. Internally, the returned set will be backed by an EnumSet. 
The iteration order of the returned set follows the enum's iteration order, not the order in
which the elements are provided to the method.
Params:
anElement – one of the elements the set should contain
otherElements – the rest of the elements the set should contain
Returns:
an immutable set containing those elements, minus duplicates/**
   * Returns an immutable set instance containing the given enum elements. Internally, the returned
   * set will be backed by an {@link EnumSet}.
   *
   * <p>The iteration order of the returned set follows the enum's iteration order, not the order in
   * which the elements are provided to the method.
   *
   * @param anElement one of the elements the set should contain
   * @param otherElements the rest of the elements the set should contain
   * @return an immutable set containing those elements, minus duplicates
   */
  // http://code.google.com/p/google-web-toolkit/issues/detail?id=3028
  @GwtCompatible(serializable = true)
  public static <E extends Enum<E>> ImmutableSet<E> immutableEnumSet(
      E anElement, E... otherElements) {
    return ImmutableEnumSet.asImmutable(EnumSet.of(anElement, otherElements));
  }

  
Returns an immutable set instance containing the given enum elements. Internally, the returned set will be backed by an EnumSet. 
The iteration order of the returned set follows the enum's iteration order, not the order in
which the elements appear in the given collection.
Params:
elements – the elements, all of the same enum type, that the set should contain
Returns:
an immutable set containing those elements, minus duplicates/**
   * Returns an immutable set instance containing the given enum elements. Internally, the returned
   * set will be backed by an {@link EnumSet}.
   *
   * <p>The iteration order of the returned set follows the enum's iteration order, not the order in
   * which the elements appear in the given collection.
   *
   * @param elements the elements, all of the same {@code enum} type, that the set should contain
   * @return an immutable set containing those elements, minus duplicates
   */
  // http://code.google.com/p/google-web-toolkit/issues/detail?id=3028
  @GwtCompatible(serializable = true)
  public static <E extends Enum<E>> ImmutableSet<E> immutableEnumSet(Iterable<E> elements) {
    if (elements instanceof ImmutableEnumSet) {
      return (ImmutableEnumSet<E>) elements;
    } else if (elements instanceof Collection) {
      Collection<E> collection = (Collection<E>) elements;
      if (collection.isEmpty()) {
        return ImmutableSet.of();
      } else {
        return ImmutableEnumSet.asImmutable(EnumSet.copyOf(collection));
      }
    } else {
      Iterator<E> itr = elements.iterator();
      if (itr.hasNext()) {
        EnumSet<E> enumSet = EnumSet.of(itr.next());
        Iterators.addAll(enumSet, itr);
        return ImmutableEnumSet.asImmutable(enumSet);
      } else {
        return ImmutableSet.of();
      }
    }
  }

  private static final class Accumulator<E extends Enum<E>> {
    static final Collector<Enum<?>, ?, ImmutableSet<? extends Enum<?>>> TO_IMMUTABLE_ENUM_SET =
        (Collector)
            Collector.<Enum, Accumulator, ImmutableSet<?>>of(
                Accumulator::new,
                Accumulator::add,
                Accumulator::combine,
                Accumulator::toImmutableSet,
                Collector.Characteristics.UNORDERED);

    @MonotonicNonNull private EnumSet<E> set;

    void add(E e) {
      if (set == null) {
        set = EnumSet.of(e);
      } else {
        set.add(e);
      }
    }

    Accumulator<E> combine(Accumulator<E> other) {
      if (this.set == null) {
        return other;
      } else if (other.set == null) {
        return this;
      } else {
        this.set.addAll(other.set);
        return this;
      }
    }

    ImmutableSet<E> toImmutableSet() {
      return (set == null) ? ImmutableSet.<E>of() : ImmutableEnumSet.asImmutable(set);
    }
  }

  
Returns a Collector that accumulates the input elements into a new ImmutableSet with an implementation specialized for enums. Unlike ImmutableSet.toImmutableSet, the resulting set will iterate over elements in their enum definition order, not encounter order. 
Since:
21.0/**
   * Returns a {@code Collector} that accumulates the input elements into a new {@code ImmutableSet}
   * with an implementation specialized for enums. Unlike {@link ImmutableSet#toImmutableSet}, the
   * resulting set will iterate over elements in their enum definition order, not encounter order.
   *
   * @since 21.0
   */
  @Beta
  public static <E extends Enum<E>> Collector<E, ?, ImmutableSet<E>> toImmutableEnumSet() {
    return (Collector) Accumulator.TO_IMMUTABLE_ENUM_SET;
  }

  
Returns a new, mutable EnumSet instance containing the given elements in their natural order. This method behaves identically to EnumSet.copyOf(Collection<Z#0-E#128>), but also accepts non-Collection iterables and empty iterables. /**
   * Returns a new, <i>mutable</i> {@code EnumSet} instance containing the given elements in their
   * natural order. This method behaves identically to {@link EnumSet#copyOf(Collection)}, but also
   * accepts non-{@code Collection} iterables and empty iterables.
   */
  public static <E extends Enum<E>> EnumSet<E> newEnumSet(
      Iterable<E> iterable, Class<E> elementType) {
    EnumSet<E> set = EnumSet.noneOf(elementType);
    Iterables.addAll(set, iterable);
    return set;
  }

  // HashSet

  
Creates a mutable, initially empty HashSet instance. 
Note: if mutability is not required, use ImmutableSet.of() instead. If 
E is an Enum type, use EnumSet.noneOf instead. Otherwise, strongly consider using a LinkedHashSet instead, at the cost of increased memory footprint, to get deterministic iteration behavior. 
Note for Java 7 and later: this method is now unnecessary and should be treated as deprecated. Instead, use the HashSet constructor directly, taking advantage of the new "diamond" syntax.
/**
   * Creates a <i>mutable</i>, initially empty {@code HashSet} instance.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSet#of()} instead. If {@code
   * E} is an {@link Enum} type, use {@link EnumSet#noneOf} instead. Otherwise, strongly consider
   * using a {@code LinkedHashSet} instead, at the cost of increased memory footprint, to get
   * deterministic iteration behavior.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code HashSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   */
  public static <E> HashSet<E> newHashSet() {
    return new HashSet<E>();
  }

  
Creates a mutable HashSet instance initially containing the given elements. 
Note: if elements are non-null and won't be added or removed after this point, use ImmutableSet.of() or ImmutableSet.copyOf(Object[]) instead. If E is an Enum type, use EnumSet.of(Enum, Enum[]) instead. Otherwise, strongly consider using a LinkedHashSet instead, at the cost of increased memory footprint, to get deterministic iteration behavior. 
This method is just a small convenience, either for newHashSet(
asList(...)), or for creating an empty set then calling Collections.addAll. This method is not actually very useful and will likely be deprecated in the future. /**
   * Creates a <i>mutable</i> {@code HashSet} instance initially containing the given elements.
   *
   * <p><b>Note:</b> if elements are non-null and won't be added or removed after this point, use
   * {@link ImmutableSet#of()} or {@link ImmutableSet#copyOf(Object[])} instead. If {@code E} is an
   * {@link Enum} type, use {@link EnumSet#of(Enum, Enum[])} instead. Otherwise, strongly consider
   * using a {@code LinkedHashSet} instead, at the cost of increased memory footprint, to get
   * deterministic iteration behavior.
   *
   * <p>This method is just a small convenience, either for {@code newHashSet(}{@link Arrays#asList
   * asList}{@code (...))}, or for creating an empty set then calling {@link Collections#addAll}.
   * This method is not actually very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(E... elements) {
    HashSet<E> set = newHashSetWithExpectedSize(elements.length);
    Collections.addAll(set, elements);
    return set;
  }

  
Creates a mutable HashSet instance containing the given elements. A very thin convenience for creating an empty set then calling Collection.addAll or Iterables.addAll. 
Note: if mutability is not required and the elements are non-null, use ImmutableSet.copyOf(Iterable<? extends Object>) instead. (Or, change elements to be a FluentIterable and call elements.toSet().) 
Note: if E is an Enum type, use newEnumSet(Iterable<Z#0-E#141>, Class<Z#0-E#141>) instead. 
Note for Java 7 and later: if elements is a Collection, you don't need this method. Instead, use the HashSet constructor directly, taking advantage of the new "diamond" syntax.
Overall, this method is not very useful and will likely be deprecated in the future.
/**
   * Creates a <i>mutable</i> {@code HashSet} instance containing the given elements. A very thin
   * convenience for creating an empty set then calling {@link Collection#addAll} or {@link
   * Iterables#addAll}.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterable)} instead. (Or, change {@code elements} to be a {@link
   * FluentIterable} and call {@code elements.toSet()}.)
   *
   * <p><b>Note:</b> if {@code E} is an {@link Enum} type, use {@link #newEnumSet(Iterable, Class)}
   * instead.
   *
   * <p><b>Note for Java 7 and later:</b> if {@code elements} is a {@link Collection}, you don't
   * need this method. Instead, use the {@code HashSet} constructor directly, taking advantage of
   * the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(Iterable<? extends E> elements) {
    return (elements instanceof Collection)
        ? new HashSet<E>(Collections2.cast(elements))
        : newHashSet(elements.iterator());
  }

  
Creates a mutable HashSet instance containing the given elements. A very thin convenience for creating an empty set and then calling Iterators.addAll. 
Note: if mutability is not required and the elements are non-null, use ImmutableSet.copyOf(Iterator<? extends Object>) instead. 
Note: if E is an Enum type, you should create an EnumSet instead. 
Overall, this method is not very useful and will likely be deprecated in the future.
/**
   * Creates a <i>mutable</i> {@code HashSet} instance containing the given elements. A very thin
   * convenience for creating an empty set and then calling {@link Iterators#addAll}.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterator)} instead.
   *
   * <p><b>Note:</b> if {@code E} is an {@link Enum} type, you should create an {@link EnumSet}
   * instead.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(Iterator<? extends E> elements) {
    HashSet<E> set = newHashSet();
    Iterators.addAll(set, elements);
    return set;
  }

  
Returns a new hash set using the smallest initial table size that can hold expectedSize elements without resizing. Note that this is not what HashSet(int) does, but it is what most users want and expect it to do. 
This behavior can't be broadly guaranteed, but has been tested with OpenJDK 1.7 and 1.8.
Params:
expectedSize – the number of elements you expect to add to the returned set
Throws:
IllegalArgumentException – if expectedSize is negative
Returns:
a new, empty hash set with enough capacity to hold expectedSize elements without resizing/**
   * Returns a new hash set using the smallest initial table size that can hold {@code expectedSize}
   * elements without resizing. Note that this is not what {@link HashSet#HashSet(int)} does, but it
   * is what most users want and expect it to do.
   *
   * <p>This behavior can't be broadly guaranteed, but has been tested with OpenJDK 1.7 and 1.8.
   *
   * @param expectedSize the number of elements you expect to add to the returned set
   * @return a new, empty hash set with enough capacity to hold {@code expectedSize} elements
   *     without resizing
   * @throws IllegalArgumentException if {@code expectedSize} is negative
   */
  public static <E> HashSet<E> newHashSetWithExpectedSize(int expectedSize) {
    return new HashSet<E>(Maps.capacity(expectedSize));
  }

  
Creates a thread-safe set backed by a hash map. The set is backed by a ConcurrentHashMap instance, and thus carries the same concurrency guarantees. 
Unlike HashSet, this class does NOT allow null to be used as an element. The set is serializable. 
Returns:
a new, empty thread-safe Set
Since:
15.0/**
   * Creates a thread-safe set backed by a hash map. The set is backed by a {@link
   * ConcurrentHashMap} instance, and thus carries the same concurrency guarantees.
   *
   * <p>Unlike {@code HashSet}, this class does NOT allow {@code null} to be used as an element. The
   * set is serializable.
   *
   * @return a new, empty thread-safe {@code Set}
   * @since 15.0
   */
  public static <E> Set<E> newConcurrentHashSet() {
    return Collections.newSetFromMap(new ConcurrentHashMap<E, Boolean>());
  }

  
Creates a thread-safe set backed by a hash map and containing the given elements. The set is backed by a ConcurrentHashMap instance, and thus carries the same concurrency guarantees. 
Unlike HashSet, this class does NOT allow null to be used as an element. The set is serializable. 
Params:
elements – the elements that the set should contain
Throws:
NullPointerException – if elements or any of its contents is null
Returns:
a new thread-safe set containing those elements (minus duplicates)
Since:
15.0/**
   * Creates a thread-safe set backed by a hash map and containing the given elements. The set is
   * backed by a {@link ConcurrentHashMap} instance, and thus carries the same concurrency
   * guarantees.
   *
   * <p>Unlike {@code HashSet}, this class does NOT allow {@code null} to be used as an element. The
   * set is serializable.
   *
   * @param elements the elements that the set should contain
   * @return a new thread-safe set containing those elements (minus duplicates)
   * @throws NullPointerException if {@code elements} or any of its contents is null
   * @since 15.0
   */
  public static <E> Set<E> newConcurrentHashSet(Iterable<? extends E> elements) {
    Set<E> set = newConcurrentHashSet();
    Iterables.addAll(set, elements);
    return set;
  }

  // LinkedHashSet

  
Creates a mutable, empty LinkedHashSet instance. 
Note: if mutability is not required, use ImmutableSet.of() instead. 
Note for Java 7 and later: this method is now unnecessary and should be treated as deprecated. Instead, use the LinkedHashSet constructor directly, taking advantage of the new "diamond" syntax.
Returns:
a new, empty LinkedHashSet/**
   * Creates a <i>mutable</i>, empty {@code LinkedHashSet} instance.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSet#of()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code LinkedHashSet} constructor directly, taking advantage of
   * the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * @return a new, empty {@code LinkedHashSet}
   */
  public static <E> LinkedHashSet<E> newLinkedHashSet() {
    return new LinkedHashSet<E>();
  }

  
Creates a mutable LinkedHashSet instance containing the given elements in order. 
Note: if mutability is not required and the elements are non-null, use ImmutableSet.copyOf(Iterable<? extends Object>) instead. 
Note for Java 7 and later: if elements is a Collection, you don't need this method. Instead, use the LinkedHashSet constructor directly, taking advantage of the new "diamond" syntax.
Overall, this method is not very useful and will likely be deprecated in the future.
Params:
elements – the elements that the set should contain, in order
Returns:
a new LinkedHashSet containing those elements (minus duplicates)/**
   * Creates a <i>mutable</i> {@code LinkedHashSet} instance containing the given elements in order.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterable)} instead.
   *
   * <p><b>Note for Java 7 and later:</b> if {@code elements} is a {@link Collection}, you don't
   * need this method. Instead, use the {@code LinkedHashSet} constructor directly, taking advantage
   * of the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   *
   * @param elements the elements that the set should contain, in order
   * @return a new {@code LinkedHashSet} containing those elements (minus duplicates)
   */
  public static <E> LinkedHashSet<E> newLinkedHashSet(Iterable<? extends E> elements) {
    if (elements instanceof Collection) {
      return new LinkedHashSet<E>(Collections2.cast(elements));
    }
    LinkedHashSet<E> set = newLinkedHashSet();
    Iterables.addAll(set, elements);
    return set;
  }

  
Creates a LinkedHashSet instance, with a high enough "initial capacity" that it should hold expectedSize elements without growth. This behavior cannot be broadly guaranteed, but it is observed to be true for OpenJDK 1.7. It also can't be guaranteed that the method isn't inadvertently oversizing the returned set.
Params:
expectedSize – the number of elements you expect to add to the returned set
Throws:
IllegalArgumentException – if expectedSize is negative
Returns:
a new, empty LinkedHashSet with enough capacity to hold expectedSize elements without resizing
Since:
11.0/**
   * Creates a {@code LinkedHashSet} instance, with a high enough "initial capacity" that it
   * <i>should</i> hold {@code expectedSize} elements without growth. This behavior cannot be
   * broadly guaranteed, but it is observed to be true for OpenJDK 1.7. It also can't be guaranteed
   * that the method isn't inadvertently <i>oversizing</i> the returned set.
   *
   * @param expectedSize the number of elements you expect to add to the returned set
   * @return a new, empty {@code LinkedHashSet} with enough capacity to hold {@code expectedSize}
   *     elements without resizing
   * @throws IllegalArgumentException if {@code expectedSize} is negative
   * @since 11.0
   */
  public static <E> LinkedHashSet<E> newLinkedHashSetWithExpectedSize(int expectedSize) {
    return new LinkedHashSet<E>(Maps.capacity(expectedSize));
  }

  // TreeSet

  
Creates a mutable, empty TreeSet instance sorted by the natural sort ordering of its elements. 
Note: if mutability is not required, use ImmutableSortedSet.of() instead. 
Note for Java 7 and later: this method is now unnecessary and should be treated as deprecated. Instead, use the TreeSet constructor directly, taking advantage of the new "diamond" syntax.
Returns:
a new, empty TreeSet/**
   * Creates a <i>mutable</i>, empty {@code TreeSet} instance sorted by the natural sort ordering of
   * its elements.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSortedSet#of()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * @return a new, empty {@code TreeSet}
   */
  public static <E extends Comparable> TreeSet<E> newTreeSet() {
    return new TreeSet<E>();
  }

  
Creates a mutable TreeSet instance containing the given elements sorted by their natural ordering. 
Note: if mutability is not required, use ImmutableSortedSet.copyOf(Iterable<? extends Object>) instead. 
Note: If elements is a SortedSet with an explicit comparator, this method has different behavior than TreeSet(SortedSet), which returns a 
TreeSet with that comparator. 
Note for Java 7 and later: this method is now unnecessary and should be treated as deprecated. Instead, use the TreeSet constructor directly, taking advantage of the new "diamond" syntax.
This method is just a small convenience for creating an empty set and then calling Iterables.addAll. This method is not very useful and will likely be deprecated in the future. 
Params:
elements – the elements that the set should contain
Returns:
a new TreeSet containing those elements (minus duplicates)/**
   * Creates a <i>mutable</i> {@code TreeSet} instance containing the given elements sorted by their
   * natural ordering.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSortedSet#copyOf(Iterable)}
   * instead.
   *
   * <p><b>Note:</b> If {@code elements} is a {@code SortedSet} with an explicit comparator, this
   * method has different behavior than {@link TreeSet#TreeSet(SortedSet)}, which returns a {@code
   * TreeSet} with that comparator.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>This method is just a small convenience for creating an empty set and then calling {@link
   * Iterables#addAll}. This method is not very useful and will likely be deprecated in the future.
   *
   * @param elements the elements that the set should contain
   * @return a new {@code TreeSet} containing those elements (minus duplicates)
   */
  public static <E extends Comparable> TreeSet<E> newTreeSet(Iterable<? extends E> elements) {
    TreeSet<E> set = newTreeSet();
    Iterables.addAll(set, elements);
    return set;
  }

  
Creates a mutable, empty TreeSet instance with the given comparator. 
Note: if mutability is not required, use 
ImmutableSortedSet.orderedBy(comparator).build() instead. 
Note for Java 7 and later: this method is now unnecessary and should be treated as deprecated. Instead, use the TreeSet constructor directly, taking advantage of the new "diamond" syntax. One caveat to this is that the 
TreeSet constructor uses a null Comparator to mean "natural ordering," whereas this factory rejects null. Clean your code accordingly. 
Params:
comparator – the comparator to use to sort the set
Throws:
NullPointerException – if comparator is null
Returns:
a new, empty TreeSet/**
   * Creates a <i>mutable</i>, empty {@code TreeSet} instance with the given comparator.
   *
   * <p><b>Note:</b> if mutability is not required, use {@code
   * ImmutableSortedSet.orderedBy(comparator).build()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>. One caveat to this is that the {@code
   * TreeSet} constructor uses a null {@code Comparator} to mean "natural ordering," whereas this
   * factory rejects null. Clean your code accordingly.
   *
   * @param comparator the comparator to use to sort the set
   * @return a new, empty {@code TreeSet}
   * @throws NullPointerException if {@code comparator} is null
   */
  public static <E> TreeSet<E> newTreeSet(Comparator<? super E> comparator) {
    return new TreeSet<E>(checkNotNull(comparator));
  }

  
Creates an empty Set that uses identity to determine equality. It compares object references, instead of calling equals, to determine whether a provided object matches an element in the set. For example, contains returns false when passed an object that equals a set member, but isn't the same instance. This behavior is similar to the way IdentityHashMap handles key lookups. 
Since:
8.0/**
   * Creates an empty {@code Set} that uses identity to determine equality. It compares object
   * references, instead of calling {@code equals}, to determine whether a provided object matches
   * an element in the set. For example, {@code contains} returns {@code false} when passed an
   * object that equals a set member, but isn't the same instance. This behavior is similar to the
   * way {@code IdentityHashMap} handles key lookups.
   *
   * @since 8.0
   */
  public static <E> Set<E> newIdentityHashSet() {
    return Collections.newSetFromMap(Maps.<E, Boolean>newIdentityHashMap());
  }

  
Creates an empty CopyOnWriteArraySet instance. 
Note: if you need an immutable empty Set, use Collections.emptySet instead. 
Returns:
a new, empty CopyOnWriteArraySet
Since:
12.0/**
   * Creates an empty {@code CopyOnWriteArraySet} instance.
   *
   * <p><b>Note:</b> if you need an immutable empty {@link Set}, use {@link Collections#emptySet}
   * instead.
   *
   * @return a new, empty {@code CopyOnWriteArraySet}
   * @since 12.0
   */
  @GwtIncompatible // CopyOnWriteArraySet
  public static <E> CopyOnWriteArraySet<E> newCopyOnWriteArraySet() {
    return new CopyOnWriteArraySet<E>();
  }

  
Creates a CopyOnWriteArraySet instance containing the given elements. 
Params:
elements – the elements that the set should contain, in order
Returns:
a new CopyOnWriteArraySet containing those elements
Since:
12.0/**
   * Creates a {@code CopyOnWriteArraySet} instance containing the given elements.
   *
   * @param elements the elements that the set should contain, in order
   * @return a new {@code CopyOnWriteArraySet} containing those elements
   * @since 12.0
   */
  @GwtIncompatible // CopyOnWriteArraySet
  public static <E> CopyOnWriteArraySet<E> newCopyOnWriteArraySet(Iterable<? extends E> elements) {
    // We copy elements to an ArrayList first, rather than incurring the
    // quadratic cost of adding them to the COWAS directly.
    Collection<? extends E> elementsCollection =
        (elements instanceof Collection)
            ? Collections2.cast(elements)
            : Lists.newArrayList(elements);
    return new CopyOnWriteArraySet<E>(elementsCollection);
  }

  
Creates an EnumSet consisting of all enum values that are not in the specified collection. If the collection is an EnumSet, this method has the same behavior as EnumSet.complementOf. Otherwise, the specified collection must contain at least one element, in order to determine the element type. If the collection could be empty, use complementOf(Collection<Z#0-E#157>, Class<Z#0-E#157>) instead of this method. 
Params:
collection – the collection whose complement should be stored in the enum set
Throws:
IllegalArgumentException – if collection is not an EnumSet instance and contains no elements
Returns:
a new, modifiable EnumSet containing all values of the enum that aren't present in the given collection/**
   * Creates an {@code EnumSet} consisting of all enum values that are not in the specified
   * collection. If the collection is an {@link EnumSet}, this method has the same behavior as
   * {@link EnumSet#complementOf}. Otherwise, the specified collection must contain at least one
   * element, in order to determine the element type. If the collection could be empty, use {@link
   * #complementOf(Collection, Class)} instead of this method.
   *
   * @param collection the collection whose complement should be stored in the enum set
   * @return a new, modifiable {@code EnumSet} containing all values of the enum that aren't present
   *     in the given collection
   * @throws IllegalArgumentException if {@code collection} is not an {@code EnumSet} instance and
   *     contains no elements
   */
  public static <E extends Enum<E>> EnumSet<E> complementOf(Collection<E> collection) {
    if (collection instanceof EnumSet) {
      return EnumSet.complementOf((EnumSet<E>) collection);
    }
    checkArgument(
        !collection.isEmpty(), "collection is empty; use the other version of this method");
    Class<E> type = collection.iterator().next().getDeclaringClass();
    return makeComplementByHand(collection, type);
  }

  
Creates an EnumSet consisting of all enum values that are not in the specified collection. This is equivalent to EnumSet.complementOf, but can act on any input collection, as long as the elements are of enum type. 
Params:
collection – the collection whose complement should be stored in the EnumSet
type – the type of the elements in the set
Returns:
a new, modifiable EnumSet initially containing all the values of the enum not present in the given collection/**
   * Creates an {@code EnumSet} consisting of all enum values that are not in the specified
   * collection. This is equivalent to {@link EnumSet#complementOf}, but can act on any input
   * collection, as long as the elements are of enum type.
   *
   * @param collection the collection whose complement should be stored in the {@code EnumSet}
   * @param type the type of the elements in the set
   * @return a new, modifiable {@code EnumSet} initially containing all the values of the enum not
   *     present in the given collection
   */
  public static <E extends Enum<E>> EnumSet<E> complementOf(
      Collection<E> collection, Class<E> type) {
    checkNotNull(collection);
    return (collection instanceof EnumSet)
        ? EnumSet.complementOf((EnumSet<E>) collection)
        : makeComplementByHand(collection, type);
  }

  private static <E extends Enum<E>> EnumSet<E> makeComplementByHand(
      Collection<E> collection, Class<E> type) {
    EnumSet<E> result = EnumSet.allOf(type);
    result.removeAll(collection);
    return result;
  }

  
Returns a set backed by the specified map. The resulting set displays the same ordering, concurrency, and performance characteristics as the backing map. In essence, this factory method provides a Set implementation corresponding to any Map implementation. There is no need to use this method on a Map implementation that already has a corresponding Set implementation (such as HashMap or TreeMap). 
Each method invocation on the set returned by this method results in exactly one method invocation on the backing map or its keySet view, with one exception. The 
addAll method is implemented as a sequence of put invocations on the backing map. 
The specified map must be empty at the time this method is invoked, and should not be
accessed directly after this method returns. These conditions are ensured if the map is created
empty, passed directly to this method, and no reference to the map is retained, as illustrated
in the following code fragment:

Set<Object> identityHashSet = Sets.newSetFromMap(
    new IdentityHashMap<Object, Boolean>());

The returned set is serializable if the backing map is.
Params:
map – the backing map
Throws:
IllegalArgumentException – if map is not empty
Returns:
the set backed by the map
Deprecated:
Use Collections.newSetFromMap instead./**
   * Returns a set backed by the specified map. The resulting set displays the same ordering,
   * concurrency, and performance characteristics as the backing map. In essence, this factory
   * method provides a {@link Set} implementation corresponding to any {@link Map} implementation.
   * There is no need to use this method on a {@link Map} implementation that already has a
   * corresponding {@link Set} implementation (such as {@link java.util.HashMap} or {@link
   * java.util.TreeMap}).
   *
   * <p>Each method invocation on the set returned by this method results in exactly one method
   * invocation on the backing map or its {@code keySet} view, with one exception. The {@code
   * addAll} method is implemented as a sequence of {@code put} invocations on the backing map.
   *
   * <p>The specified map must be empty at the time this method is invoked, and should not be
   * accessed directly after this method returns. These conditions are ensured if the map is created
   * empty, passed directly to this method, and no reference to the map is retained, as illustrated
   * in the following code fragment:
   *
   * <pre>{@code
   * Set<Object> identityHashSet = Sets.newSetFromMap(
   *     new IdentityHashMap<Object, Boolean>());
   * }</pre>
   *
   * <p>The returned set is serializable if the backing map is.
   *
   * @param map the backing map
   * @return the set backed by the map
   * @throws IllegalArgumentException if {@code map} is not empty
   * @deprecated Use {@link Collections#newSetFromMap} instead.
   */
  @Deprecated
  public static <E> Set<E> newSetFromMap(Map<E, Boolean> map) {
    return Collections.newSetFromMap(map);
  }

  
An unmodifiable view of a set which may be backed by other sets; this view will change as the backing sets do. Contains methods to copy the data into a new set which will then remain stable. There is usually no reason to retain a reference of type SetView; typically, you either use it as a plain Set, or immediately invoke immutableCopy or copyInto and forget the SetView itself. 
Since:
2.0/**
   * An unmodifiable view of a set which may be backed by other sets; this view will change as the
   * backing sets do. Contains methods to copy the data into a new set which will then remain
   * stable. There is usually no reason to retain a reference of type {@code SetView}; typically,
   * you either use it as a plain {@link Set}, or immediately invoke {@link #immutableCopy} or
   * {@link #copyInto} and forget the {@code SetView} itself.
   *
   * @since 2.0
   */
  public abstract static class SetView<E> extends AbstractSet<E> {
    private SetView() {} // no subclasses but our own

    
Returns an immutable copy of the current contents of this set view. Does not support null
elements.
Warning: this may have unexpected results if a backing set of this view uses a nonstandard notion of equivalence, for example if it is a TreeSet using a comparator that is inconsistent with Object.equals(Object). /**
     * Returns an immutable copy of the current contents of this set view. Does not support null
     * elements.
     *
     * <p><b>Warning:</b> this may have unexpected results if a backing set of this view uses a
     * nonstandard notion of equivalence, for example if it is a {@link TreeSet} using a comparator
     * that is inconsistent with {@link Object#equals(Object)}.
     */
    public ImmutableSet<E> immutableCopy() {
      return ImmutableSet.copyOf(this);
    }

    
Copies the current contents of this set view into an existing set. This method has equivalent behavior to set.addAll(this), assuming that all the sets involved are based on the same notion of equivalence. 
Returns:
a reference to set, for convenience/**
     * Copies the current contents of this set view into an existing set. This method has equivalent
     * behavior to {@code set.addAll(this)}, assuming that all the sets involved are based on the
     * same notion of equivalence.
     *
     * @return a reference to {@code set}, for convenience
     */
    // Note: S should logically extend Set<? super E> but can't due to either
    // some javac bug or some weirdness in the spec, not sure which.
    @CanIgnoreReturnValue
    public <S extends Set<E>> S copyInto(S set) {
      set.addAll(this);
      return set;
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean add(E e) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean remove(Object object) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean addAll(Collection<? extends E> newElements) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean removeAll(Collection<?> oldElements) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean removeIf(java.util.function.Predicate<? super E> filter) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean retainAll(Collection<?> elementsToKeep) {
      throw new UnsupportedOperationException();
    }

    
Guaranteed to throw an exception and leave the collection unmodified.
Throws:
UnsupportedOperationException – always
Deprecated:
Unsupported operation./**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @Deprecated
    @Override
    public final void clear() {
      throw new UnsupportedOperationException();
    }

    
Scope the return type to UnmodifiableIterator to ensure this is an unmodifiable view. 
Since:
20.0 (present with return type Iterator since 2.0)/**
     * Scope the return type to {@link UnmodifiableIterator} to ensure this is an unmodifiable view.
     *
     * @since 20.0 (present with return type {@link Iterator} since 2.0)
     */
    @Override
    public abstract UnmodifiableIterator<E> iterator();
  }

  
Returns an unmodifiable view of the union of two sets. The returned set contains all elements that are contained in either backing set. Iterating over the returned set iterates first over all the elements of set1, then over each element of set2, in order, that is not contained in set1. 
Results are undefined if set1 and set2 are sets based on different equivalence relations (as HashSet, TreeSet, and the Map.keySet of an IdentityHashMap all are). /**
   * Returns an unmodifiable <b>view</b> of the union of two sets. The returned set contains all
   * elements that are contained in either backing set. Iterating over the returned set iterates
   * first over all the elements of {@code set1}, then over each element of {@code set2}, in order,
   * that is not contained in {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@link HashSet}, {@link TreeSet}, and the {@link Map#keySet} of an
   * {@code IdentityHashMap} all are).
   */
  public static <E> SetView<E> union(final Set<? extends E> set1, final Set<? extends E> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public int size() {
        int size = set1.size();
        for (E e : set2) {
          if (!set1.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set1.isEmpty() && set2.isEmpty();
      }

      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<? extends E> itr1 = set1.iterator();
          final Iterator<? extends E> itr2 = set2.iterator();

          @Override
          protected E computeNext() {
            if (itr1.hasNext()) {
              return itr1.next();
            }
            while (itr2.hasNext()) {
              E e = itr2.next();
              if (!set1.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public Stream<E> stream() {
        return Stream.concat(set1.stream(), set2.stream().filter(e -> !set1.contains(e)));
      }

      @Override
      public Stream<E> parallelStream() {
        return stream().parallel();
      }

      @Override
      public boolean contains(Object object) {
        return set1.contains(object) || set2.contains(object);
      }

      @Override
      public <S extends Set<E>> S copyInto(S set) {
        set.addAll(set1);
        set.addAll(set2);
        return set;
      }

      @Override
      public ImmutableSet<E> immutableCopy() {
        return new ImmutableSet.Builder<E>().addAll(set1).addAll(set2).build();
      }
    };
  }

  
Returns an unmodifiable view of the intersection of two sets. The returned set contains all elements that are contained by both backing sets. The iteration order of the returned set matches that of set1. 
Results are undefined if set1 and set2 are sets based on different equivalence relations (as HashSet, TreeSet, and the keySet of an 
IdentityHashMap all are). 
Note: The returned view performs slightly better when set1 is the smaller of the two sets. If you have reason to believe one of your sets will generally be smaller than the other, pass it first. Unfortunately, since this method sets the generic type of the returned set based on the type of the first set passed, this could in rare cases force you to make a cast, for example: 

Set<Object> aFewBadObjects = ...
Set<String> manyBadStrings = ...
// impossible for a non-String to be in the intersection
SuppressWarnings("unchecked")
Set<String> badStrings = (Set) Sets.intersection(
    aFewBadObjects, manyBadStrings);

This is unfortunate, but should come up only very rarely.
/**
   * Returns an unmodifiable <b>view</b> of the intersection of two sets. The returned set contains
   * all elements that are contained by both backing sets. The iteration order of the returned set
   * matches that of {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   *
   * <p><b>Note:</b> The returned view performs slightly better when {@code set1} is the smaller of
   * the two sets. If you have reason to believe one of your sets will generally be smaller than the
   * other, pass it first. Unfortunately, since this method sets the generic type of the returned
   * set based on the type of the first set passed, this could in rare cases force you to make a
   * cast, for example:
   *
   * <pre>{@code
   * Set<Object> aFewBadObjects = ...
   * Set<String> manyBadStrings = ...
   *
   * // impossible for a non-String to be in the intersection
   * SuppressWarnings("unchecked")
   * Set<String> badStrings = (Set) Sets.intersection(
   *     aFewBadObjects, manyBadStrings);
   * }</pre>
   *
   * <p>This is unfortunate, but should come up only very rarely.
   */
  public static <E> SetView<E> intersection(final Set<E> set1, final Set<?> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<E> itr = set1.iterator();

          @Override
          protected E computeNext() {
            while (itr.hasNext()) {
              E e = itr.next();
              if (set2.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public Stream<E> stream() {
        return set1.stream().filter(set2::contains);
      }

      @Override
      public Stream<E> parallelStream() {
        return set1.parallelStream().filter(set2::contains);
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (set2.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return Collections.disjoint(set1, set2);
      }

      @Override
      public boolean contains(Object object) {
        return set1.contains(object) && set2.contains(object);
      }

      @Override
      public boolean containsAll(Collection<?> collection) {
        return set1.containsAll(collection) && set2.containsAll(collection);
      }
    };
  }

  
Returns an unmodifiable view of the difference of two sets. The returned set contains all elements that are contained by set1 and not contained by set2. set2 may also contain elements not present in set1; these are simply ignored. The iteration order of the returned set matches that of set1. 
Results are undefined if set1 and set2 are sets based on different equivalence relations (as HashSet, TreeSet, and the keySet of an 
IdentityHashMap all are). /**
   * Returns an unmodifiable <b>view</b> of the difference of two sets. The returned set contains
   * all elements that are contained by {@code set1} and not contained by {@code set2}. {@code set2}
   * may also contain elements not present in {@code set1}; these are simply ignored. The iteration
   * order of the returned set matches that of {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   */
  public static <E> SetView<E> difference(final Set<E> set1, final Set<?> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<E> itr = set1.iterator();

          @Override
          protected E computeNext() {
            while (itr.hasNext()) {
              E e = itr.next();
              if (!set2.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public Stream<E> stream() {
        return set1.stream().filter(e -> !set2.contains(e));
      }

      @Override
      public Stream<E> parallelStream() {
        return set1.parallelStream().filter(e -> !set2.contains(e));
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (!set2.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set2.containsAll(set1);
      }

      @Override
      public boolean contains(Object element) {
        return set1.contains(element) && !set2.contains(element);
      }
    };
  }

  
Returns an unmodifiable view of the symmetric difference of two sets. The returned set contains all elements that are contained in either set1 or set2 but not in both. The iteration order of the returned set is undefined. 
Results are undefined if set1 and set2 are sets based on different equivalence relations (as HashSet, TreeSet, and the keySet of an 
IdentityHashMap all are). 
Since:
3.0/**
   * Returns an unmodifiable <b>view</b> of the symmetric difference of two sets. The returned set
   * contains all elements that are contained in either {@code set1} or {@code set2} but not in
   * both. The iteration order of the returned set is undefined.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   *
   * @since 3.0
   */
  public static <E> SetView<E> symmetricDifference(
      final Set<? extends E> set1, final Set<? extends E> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        final Iterator<? extends E> itr1 = set1.iterator();
        final Iterator<? extends E> itr2 = set2.iterator();
        return new AbstractIterator<E>() {
          @Override
          public E computeNext() {
            while (itr1.hasNext()) {
              E elem1 = itr1.next();
              if (!set2.contains(elem1)) {
                return elem1;
              }
            }
            while (itr2.hasNext()) {
              E elem2 = itr2.next();
              if (!set1.contains(elem2)) {
                return elem2;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (!set2.contains(e)) {
            size++;
          }
        }
        for (E e : set2) {
          if (!set1.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set1.equals(set2);
      }

      @Override
      public boolean contains(Object element) {
        return set1.contains(element) ^ set2.contains(element);
      }
    };
  }

  
Returns the elements of unfiltered that satisfy a predicate. The returned set is a live view of unfiltered; changes to one affect the other. 
The resulting set's iterator does not support remove(), but all other set methods are supported. When given an element that doesn't satisfy the predicate, the set's 
add() and addAll() methods throw an IllegalArgumentException. When methods such as removeAll() and clear() are called on the filtered set, only elements that satisfy the filter will be removed from the underlying set. 
The returned set isn't threadsafe or serializable, even if unfiltered is. 
Many of the filtered set's methods, such as size(), iterate across every element in the underlying set and determine which elements satisfy the filter. When a live view is not needed, it may be faster to copy Iterables.filter(unfiltered, predicate) and use the copy. 
Warning: predicate must be consistent with equals, as documented at Predicate.apply. Do not provide a predicate such as 
Predicates.instanceOf(ArrayList.class), which is inconsistent with equals. (See Iterables.filter(Iterable<?>, Class<Object>) for related functionality.) 
Java 8 users: many use cases for this method are better addressed by Stream.filter. This method is not being deprecated, but we gently encourage you to migrate to streams. /**
   * Returns the elements of {@code unfiltered} that satisfy a predicate. The returned set is a live
   * view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * <p><b>Java 8 users:</b> many use cases for this method are better addressed by {@link
   * java.util.stream.Stream#filter}. This method is not being deprecated, but we gently encourage
   * you to migrate to streams.
   */
  // TODO(kevinb): how to omit that last sentence when building GWT javadoc?
  public static <E> Set<E> filter(Set<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof SortedSet) {
      return filter((SortedSet<E>) unfiltered, predicate);
    }
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredSet<E>((Set<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  
Returns the elements of a SortedSet, unfiltered, that satisfy a predicate. The returned set is a live view of unfiltered; changes to one affect the other. 
The resulting set's iterator does not support remove(), but all other set methods are supported. When given an element that doesn't satisfy the predicate, the set's 
add() and addAll() methods throw an IllegalArgumentException. When methods such as removeAll() and clear() are called on the filtered set, only elements that satisfy the filter will be removed from the underlying set. 
The returned set isn't threadsafe or serializable, even if unfiltered is. 
Many of the filtered set's methods, such as size(), iterate across every element in the underlying set and determine which elements satisfy the filter. When a live view is not needed, it may be faster to copy Iterables.filter(unfiltered, predicate) and use the copy. 
Warning: predicate must be consistent with equals, as documented at Predicate.apply. Do not provide a predicate such as 
Predicates.instanceOf(ArrayList.class), which is inconsistent with equals. (See Iterables.filter(Iterable<?>, Class<Object>) for related functionality.) 
Since:
11.0/**
   * Returns the elements of a {@code SortedSet}, {@code unfiltered}, that satisfy a predicate. The
   * returned set is a live view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * @since 11.0
   */
  public static <E> SortedSet<E> filter(SortedSet<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredSortedSet<E>((SortedSet<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredSortedSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  
Returns the elements of a NavigableSet, unfiltered, that satisfy a predicate. The returned set is a live view of unfiltered; changes to one affect the other. 
The resulting set's iterator does not support remove(), but all other set methods are supported. When given an element that doesn't satisfy the predicate, the set's 
add() and addAll() methods throw an IllegalArgumentException. When methods such as removeAll() and clear() are called on the filtered set, only elements that satisfy the filter will be removed from the underlying set. 
The returned set isn't threadsafe or serializable, even if unfiltered is. 
Many of the filtered set's methods, such as size(), iterate across every element in the underlying set and determine which elements satisfy the filter. When a live view is not needed, it may be faster to copy Iterables.filter(unfiltered, predicate) and use the copy. 
Warning: predicate must be consistent with equals, as documented at Predicate.apply. Do not provide a predicate such as 
Predicates.instanceOf(ArrayList.class), which is inconsistent with equals. (See Iterables.filter(Iterable<?>, Class<Object>) for related functionality.) 
Since:
14.0/**
   * Returns the elements of a {@code NavigableSet}, {@code unfiltered}, that satisfy a predicate.
   * The returned set is a live view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * @since 14.0
   */
  @GwtIncompatible // NavigableSet
  @SuppressWarnings("unchecked")
  public static <E> NavigableSet<E> filter(
      NavigableSet<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredNavigableSet<E>((NavigableSet<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredNavigableSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  private static class FilteredSet<E> extends FilteredCollection<E> implements Set<E> {
    FilteredSet(Set<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    @Override
    public boolean equals(@Nullable Object object) {
      return equalsImpl(this, object);
    }

    @Override
    public int hashCode() {
      return hashCodeImpl(this);
    }
  }

  private static class FilteredSortedSet<E> extends FilteredSet<E> implements SortedSet<E> {

    FilteredSortedSet(SortedSet<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    @Override
    public Comparator<? super E> comparator() {
      return ((SortedSet<E>) unfiltered).comparator();
    }

    @Override
    public SortedSet<E> subSet(E fromElement, E toElement) {
      return new FilteredSortedSet<E>(
          ((SortedSet<E>) unfiltered).subSet(fromElement, toElement), predicate);
    }

    @Override
    public SortedSet<E> headSet(E toElement) {
      return new FilteredSortedSet<E>(((SortedSet<E>) unfiltered).headSet(toElement), predicate);
    }

    @Override
    public SortedSet<E> tailSet(E fromElement) {
      return new FilteredSortedSet<E>(((SortedSet<E>) unfiltered).tailSet(fromElement), predicate);
    }

    @Override
    public E first() {
      return Iterators.find(unfiltered.iterator(), predicate);
    }

    @Override
    public E last() {
      SortedSet<E> sortedUnfiltered = (SortedSet<E>) unfiltered;
      while (true) {
        E element = sortedUnfiltered.last();
        if (predicate.apply(element)) {
          return element;
        }
        sortedUnfiltered = sortedUnfiltered.headSet(element);
      }
    }
  }

  @GwtIncompatible // NavigableSet
  private static class FilteredNavigableSet<E> extends FilteredSortedSet<E>
      implements NavigableSet<E> {
    FilteredNavigableSet(NavigableSet<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    NavigableSet<E> unfiltered() {
      return (NavigableSet<E>) unfiltered;
    }

    @Override
    public @Nullable E lower(E e) {
      return Iterators.find(unfiltered().headSet(e, false).descendingIterator(), predicate, null);
    }

    @Override
    public @Nullable E floor(E e) {
      return Iterators.find(unfiltered().headSet(e, true).descendingIterator(), predicate, null);
    }

    @Override
    public E ceiling(E e) {
      return Iterables.find(unfiltered().tailSet(e, true), predicate, null);
    }

    @Override
    public E higher(E e) {
      return Iterables.find(unfiltered().tailSet(e, false), predicate, null);
    }

    @Override
    public E pollFirst() {
      return Iterables.removeFirstMatching(unfiltered(), predicate);
    }

    @Override
    public E pollLast() {
      return Iterables.removeFirstMatching(unfiltered().descendingSet(), predicate);
    }

    @Override
    public NavigableSet<E> descendingSet() {
      return Sets.filter(unfiltered().descendingSet(), predicate);
    }

    @Override
    public Iterator<E> descendingIterator() {
      return Iterators.filter(unfiltered().descendingIterator(), predicate);
    }

    @Override
    public E last() {
      return Iterators.find(unfiltered().descendingIterator(), predicate);
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return filter(
          unfiltered().subSet(fromElement, fromInclusive, toElement, toInclusive), predicate);
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return filter(unfiltered().headSet(toElement, inclusive), predicate);
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return filter(unfiltered().tailSet(fromElement, inclusive), predicate);
    }
  }

  
Returns every possible list that can be formed by choosing one element from each of the given
sets in order; the "n-ary Cartesian
product" of the sets. For example:

Sets.cartesianProduct(ImmutableList.of(
    ImmutableSet.of(1, 2),
    ImmutableSet.of("A", "B", "C")))

returns a set containing six lists:

  
ImmutableList.of(1, "A") 
ImmutableList.of(1, "B") 
ImmutableList.of(1, "C") 
ImmutableList.of(2, "A") 
ImmutableList.of(2, "B") 
ImmutableList.of(2, "C") 
The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
products that you would get from nesting for loops:

for (B b0 : sets.get(0)) {
  for (B b1 : sets.get(1)) {
    ...
    ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
    // operate on tuple
  }
 }

Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
all are provided (an empty list), the resulting Cartesian product has one element, an empty
list (counter-intuitive, but mathematically consistent).
Performance notes: while the cartesian product of sets of size m, n, p is a set of size m x n x p, its actual memory consumption is much smaller. When the cartesian set is constructed, the input sets are merely copied. Only as the resulting set is iterated are the individual lists created, and these are not retained after iteration. 
Params:
sets – the sets to choose elements from, in the order that the elements chosen from those
    sets should appear in the resulting lists
Type parameters:
<B> – any common base class shared by all axes (often just Object)
Throws:
NullPointerException – if sets, any one of the sets, or any element of a provided set is null
Returns:
the Cartesian product, as an immutable set containing immutable lists
Since:
2.0/**
   * Returns every possible list that can be formed by choosing one element from each of the given
   * sets in order; the "n-ary <a href="http://en.wikipedia.org/wiki/Cartesian_product">Cartesian
   * product</a>" of the sets. For example:
   *
   * <pre>{@code
   * Sets.cartesianProduct(ImmutableList.of(
   *     ImmutableSet.of(1, 2),
   *     ImmutableSet.of("A", "B", "C")))
   * }</pre>
   *
   * <p>returns a set containing six lists:
   *
   * <ul>
   *   <li>{@code ImmutableList.of(1, "A")}
   *   <li>{@code ImmutableList.of(1, "B")}
   *   <li>{@code ImmutableList.of(1, "C")}
   *   <li>{@code ImmutableList.of(2, "A")}
   *   <li>{@code ImmutableList.of(2, "B")}
   *   <li>{@code ImmutableList.of(2, "C")}
   * </ul>
   *
   * <p>The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
   * products that you would get from nesting for loops:
   *
   * <pre>{@code
   * for (B b0 : sets.get(0)) {
   *   for (B b1 : sets.get(1)) {
   *     ...
   *     ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
   *     // operate on tuple
   *   }
   * }
   * }</pre>
   *
   * <p>Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
   * all are provided (an empty list), the resulting Cartesian product has one element, an empty
   * list (counter-intuitive, but mathematically consistent).
   *
   * <p><i>Performance notes:</i> while the cartesian product of sets of size {@code m, n, p} is a
   * set of size {@code m x n x p}, its actual memory consumption is much smaller. When the
   * cartesian set is constructed, the input sets are merely copied. Only as the resulting set is
   * iterated are the individual lists created, and these are not retained after iteration.
   *
   * @param sets the sets to choose elements from, in the order that the elements chosen from those
   *     sets should appear in the resulting lists
   * @param <B> any common base class shared by all axes (often just {@link Object})
   * @return the Cartesian product, as an immutable set containing immutable lists
   * @throws NullPointerException if {@code sets}, any one of the {@code sets}, or any element of a
   *     provided set is null
   * @since 2.0
   */
  public static <B> Set<List<B>> cartesianProduct(List<? extends Set<? extends B>> sets) {
    return CartesianSet.create(sets);
  }

  
Returns every possible list that can be formed by choosing one element from each of the given
sets in order; the "n-ary Cartesian
product" of the sets. For example:

Sets.cartesianProduct(
    ImmutableSet.of(1, 2),
    ImmutableSet.of("A", "B", "C"))

returns a set containing six lists:

  
ImmutableList.of(1, "A") 
ImmutableList.of(1, "B") 
ImmutableList.of(1, "C") 
ImmutableList.of(2, "A") 
ImmutableList.of(2, "B") 
ImmutableList.of(2, "C") 
The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
products that you would get from nesting for loops:

for (B b0 : sets.get(0)) {
  for (B b1 : sets.get(1)) {
    ...
    ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
    // operate on tuple
  }
 }

Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
all are provided (an empty list), the resulting Cartesian product has one element, an empty
list (counter-intuitive, but mathematically consistent).
Performance notes: while the cartesian product of sets of size m, n, p is a set of size m x n x p, its actual memory consumption is much smaller. When the cartesian set is constructed, the input sets are merely copied. Only as the resulting set is iterated are the individual lists created, and these are not retained after iteration. 
Params:
sets – the sets to choose elements from, in the order that the elements chosen from those
    sets should appear in the resulting lists
Type parameters:
<B> – any common base class shared by all axes (often just Object)
Throws:
NullPointerException – if sets, any one of the sets, or any element of a provided set is null
Returns:
the Cartesian product, as an immutable set containing immutable lists
Since:
2.0/**
   * Returns every possible list that can be formed by choosing one element from each of the given
   * sets in order; the "n-ary <a href="http://en.wikipedia.org/wiki/Cartesian_product">Cartesian
   * product</a>" of the sets. For example:
   *
   * <pre>{@code
   * Sets.cartesianProduct(
   *     ImmutableSet.of(1, 2),
   *     ImmutableSet.of("A", "B", "C"))
   * }</pre>
   *
   * <p>returns a set containing six lists:
   *
   * <ul>
   *   <li>{@code ImmutableList.of(1, "A")}
   *   <li>{@code ImmutableList.of(1, "B")}
   *   <li>{@code ImmutableList.of(1, "C")}
   *   <li>{@code ImmutableList.of(2, "A")}
   *   <li>{@code ImmutableList.of(2, "B")}
   *   <li>{@code ImmutableList.of(2, "C")}
   * </ul>
   *
   * <p>The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
   * products that you would get from nesting for loops:
   *
   * <pre>{@code
   * for (B b0 : sets.get(0)) {
   *   for (B b1 : sets.get(1)) {
   *     ...
   *     ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
   *     // operate on tuple
   *   }
   * }
   * }</pre>
   *
   * <p>Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
   * all are provided (an empty list), the resulting Cartesian product has one element, an empty
   * list (counter-intuitive, but mathematically consistent).
   *
   * <p><i>Performance notes:</i> while the cartesian product of sets of size {@code m, n, p} is a
   * set of size {@code m x n x p}, its actual memory consumption is much smaller. When the
   * cartesian set is constructed, the input sets are merely copied. Only as the resulting set is
   * iterated are the individual lists created, and these are not retained after iteration.
   *
   * @param sets the sets to choose elements from, in the order that the elements chosen from those
   *     sets should appear in the resulting lists
   * @param <B> any common base class shared by all axes (often just {@link Object})
   * @return the Cartesian product, as an immutable set containing immutable lists
   * @throws NullPointerException if {@code sets}, any one of the {@code sets}, or any element of a
   *     provided set is null
   * @since 2.0
   */
  @SafeVarargs
  public static <B> Set<List<B>> cartesianProduct(Set<? extends B>... sets) {
    return cartesianProduct(Arrays.asList(sets));
  }

  private static final class CartesianSet<E> extends ForwardingCollection<List<E>>
      implements Set<List<E>> {
    private final transient ImmutableList<ImmutableSet<E>> axes;
    private final transient CartesianList<E> delegate;

    static <E> Set<List<E>> create(List<? extends Set<? extends E>> sets) {
      ImmutableList.Builder<ImmutableSet<E>> axesBuilder = new ImmutableList.Builder<>(sets.size());
      for (Set<? extends E> set : sets) {
        ImmutableSet<E> copy = ImmutableSet.copyOf(set);
        if (copy.isEmpty()) {
          return ImmutableSet.of();
        }
        axesBuilder.add(copy);
      }
      final ImmutableList<ImmutableSet<E>> axes = axesBuilder.build();
      ImmutableList<List<E>> listAxes =
          new ImmutableList<List<E>>() {
            @Override
            public int size() {
              return axes.size();
            }

            @Override
            public List<E> get(int index) {
              return axes.get(index).asList();
            }

            @Override
            boolean isPartialView() {
              return true;
            }
          };
      return new CartesianSet<E>(axes, new CartesianList<E>(listAxes));
    }

    private CartesianSet(ImmutableList<ImmutableSet<E>> axes, CartesianList<E> delegate) {
      this.axes = axes;
      this.delegate = delegate;
    }

    @Override
    protected Collection<List<E>> delegate() {
      return delegate;
    }

    @Override
    public boolean equals(@Nullable Object object) {
      // Warning: this is broken if size() == 0, so it is critical that we
      // substitute an empty ImmutableSet to the user in place of this
      if (object instanceof CartesianSet) {
        CartesianSet<?> that = (CartesianSet<?>) object;
        return this.axes.equals(that.axes);
      }
      return super.equals(object);
    }

    @Override
    public int hashCode() {
      // Warning: this is broken if size() == 0, so it is critical that we
      // substitute an empty ImmutableSet to the user in place of this

      // It's a weird formula, but tests prove it works.
      int adjust = size() - 1;
      for (int i = 0; i < axes.size(); i++) {
        adjust *= 31;
        adjust = ~~adjust;
        // in GWT, we have to deal with integer overflow carefully
      }
      int hash = 1;
      for (Set<E> axis : axes) {
        hash = 31 * hash + (size() / axis.size() * axis.hashCode());

        hash = ~~hash;
      }
      hash += adjust;
      return ~~hash;
    }
  }

  
Returns the set of all possible subsets of set. For example, 
powerSet(ImmutableSet.of(1, 2)) returns the set {{}, {1}, {2}, {1, 2}}. 
Elements appear in these subsets in the same iteration order as they appeared in the input
set. The order in which these subsets appear in the outer set is undefined. Note that the power
set of the empty set is not the empty set, but a one-element set containing the empty set.
The returned set and its constituent sets use equals to decide whether two elements are identical, even if the input set uses a different concept of equivalence. 
Performance notes: while the power set of a set with size n is of size 
2^n, its memory usage is only O(n). When the power set is constructed, the input set is merely copied. Only as the power set is iterated are the individual subsets created, and these subsets themselves occupy only a small constant amount of memory. 
Params:
set – the set of elements to construct a power set from
Throws:
IllegalArgumentException – if set has more than 30 unique elements (causing the power set size to exceed the int range)
NullPointerException – if set is or contains null
See Also:
Power set article at Wikipedia
Returns:
the power set, as an immutable set of immutable sets
Since:
4.0/**
   * Returns the set of all possible subsets of {@code set}. For example, {@code
   * powerSet(ImmutableSet.of(1, 2))} returns the set {@code {{}, {1}, {2}, {1, 2}}}.
   *
   * <p>Elements appear in these subsets in the same iteration order as they appeared in the input
   * set. The order in which these subsets appear in the outer set is undefined. Note that the power
   * set of the empty set is not the empty set, but a one-element set containing the empty set.
   *
   * <p>The returned set and its constituent sets use {@code equals} to decide whether two elements
   * are identical, even if the input set uses a different concept of equivalence.
   *
   * <p><i>Performance notes:</i> while the power set of a set with size {@code n} is of size {@code
   * 2^n}, its memory usage is only {@code O(n)}. When the power set is constructed, the input set
   * is merely copied. Only as the power set is iterated are the individual subsets created, and
   * these subsets themselves occupy only a small constant amount of memory.
   *
   * @param set the set of elements to construct a power set from
   * @return the power set, as an immutable set of immutable sets
   * @throws IllegalArgumentException if {@code set} has more than 30 unique elements (causing the
   *     power set size to exceed the {@code int} range)
   * @throws NullPointerException if {@code set} is or contains {@code null}
   * @see <a href="http://en.wikipedia.org/wiki/Power_set">Power set article at Wikipedia</a>
   * @since 4.0
   */
  @GwtCompatible(serializable = false)
  public static <E> Set<Set<E>> powerSet(Set<E> set) {
    return new PowerSet<E>(set);
  }

  private static final class SubSet<E> extends AbstractSet<E> {
    private final ImmutableMap<E, Integer> inputSet;
    private final int mask;

    SubSet(ImmutableMap<E, Integer> inputSet, int mask) {
      this.inputSet = inputSet;
      this.mask = mask;
    }

    @Override
    public Iterator<E> iterator() {
      return new UnmodifiableIterator<E>() {
        final ImmutableList<E> elements = inputSet.keySet().asList();
        int remainingSetBits = mask;

        @Override
        public boolean hasNext() {
          return remainingSetBits != 0;
        }

        @Override
        public E next() {
          int index = Integer.numberOfTrailingZeros(remainingSetBits);
          if (index == 32) {
            throw new NoSuchElementException();
          }
          remainingSetBits &= ~(1 << index);
          return elements.get(index);
        }
      };
    }

    @Override
    public int size() {
      return Integer.bitCount(mask);
    }

    @Override
    public boolean contains(@Nullable Object o) {
      Integer index = inputSet.get(o);
      return index != null && (mask & (1 << index)) != 0;
    }
  }

  private static final class PowerSet<E> extends AbstractSet<Set<E>> {
    final ImmutableMap<E, Integer> inputSet;

    PowerSet(Set<E> input) {
      this.inputSet = Maps.indexMap(input);
      checkArgument(
          inputSet.size() <= 30, "Too many elements to create power set: %s > 30", inputSet.size());
    }

    @Override
    public int size() {
      return 1 << inputSet.size();
    }

    @Override
    public boolean isEmpty() {
      return false;
    }

    @Override
    public Iterator<Set<E>> iterator() {
      return new AbstractIndexedListIterator<Set<E>>(size()) {
        @Override
        protected Set<E> get(final int setBits) {
          return new SubSet<E>(inputSet, setBits);
        }
      };
    }

    @Override
    public boolean contains(@Nullable Object obj) {
      if (obj instanceof Set) {
        Set<?> set = (Set<?>) obj;
        return inputSet.keySet().containsAll(set);
      }
      return false;
    }

    @Override
    public boolean equals(@Nullable Object obj) {
      if (obj instanceof PowerSet) {
        PowerSet<?> that = (PowerSet<?>) obj;
        return inputSet.equals(that.inputSet);
      }
      return super.equals(obj);
    }

    @Override
    public int hashCode() {
      /*
       * The sum of the sums of the hash codes in each subset is just the sum of
       * each input element's hash code times the number of sets that element
       * appears in. Each element appears in exactly half of the 2^n sets, so:
       */
      return inputSet.keySet().hashCode() << (inputSet.size() - 1);
    }

    @Override
    public String toString() {
      return "powerSet(" + inputSet + ")";
    }
  }

  
Returns the set of all subsets of set of size size. For example, 
combinations(ImmutableSet.of(1, 2, 3), 2) returns the set {{1, 2}, {1, 3}, {2, 3}}. 
Elements appear in these subsets in the same iteration order as they appeared in the input
set. The order in which these subsets appear in the outer set is undefined.
The returned set and its constituent sets use equals to decide whether two elements are identical, even if the input set uses a different concept of equivalence. 
Performance notes: the memory usage of the returned set is only O(n). When the result set is constructed, the input set is merely copied. Only as the result set is iterated are the individual subsets created. Each of these subsets occupies an additional O(n) memory but only for as long as the user retains a reference to it. That is, the set returned by combinations does not retain the individual subsets. 
Params:
set – the set of elements to take combinations of
size – the number of elements per combination
Throws:
IllegalArgumentException – if size is not between 0 and set.size() inclusive
NullPointerException – if set is or contains null
Returns:
the set of all combinations of size elements from set
Since:
23.0/**
   * Returns the set of all subsets of {@code set} of size {@code size}. For example, {@code
   * combinations(ImmutableSet.of(1, 2, 3), 2)} returns the set {@code {{1, 2}, {1, 3}, {2, 3}}}.
   *
   * <p>Elements appear in these subsets in the same iteration order as they appeared in the input
   * set. The order in which these subsets appear in the outer set is undefined.
   *
   * <p>The returned set and its constituent sets use {@code equals} to decide whether two elements
   * are identical, even if the input set uses a different concept of equivalence.
   *
   * <p><i>Performance notes:</i> the memory usage of the returned set is only {@code O(n)}. When
   * the result set is constructed, the input set is merely copied. Only as the result set is
   * iterated are the individual subsets created. Each of these subsets occupies an additional O(n)
   * memory but only for as long as the user retains a reference to it. That is, the set returned by
   * {@code combinations} does not retain the individual subsets.
   *
   * @param set the set of elements to take combinations of
   * @param size the number of elements per combination
   * @return the set of all combinations of {@code size} elements from {@code set}
   * @throws IllegalArgumentException if {@code size} is not between 0 and {@code set.size()}
   *     inclusive
   * @throws NullPointerException if {@code set} is or contains {@code null}
   * @since 23.0
   */
  @Beta
  public static <E> Set<Set<E>> combinations(Set<E> set, final int size) {
    final ImmutableMap<E, Integer> index = Maps.indexMap(set);
    checkNonnegative(size, "size");
    checkArgument(size <= index.size(), "size (%s) must be <= set.size() (%s)", size, index.size());
    if (size == 0) {
      return ImmutableSet.<Set<E>>of(ImmutableSet.<E>of());
    } else if (size == index.size()) {
      return ImmutableSet.<Set<E>>of(index.keySet());
    }
    return new AbstractSet<Set<E>>() {
      @Override
      public boolean contains(@Nullable Object o) {
        if (o instanceof Set) {
          Set<?> s = (Set<?>) o;
          return s.size() == size && index.keySet().containsAll(s);
        }
        return false;
      }

      @Override
      public Iterator<Set<E>> iterator() {
        return new AbstractIterator<Set<E>>() {
          final BitSet bits = new BitSet(index.size());

          @Override
          protected Set<E> computeNext() {
            if (bits.isEmpty()) {
              bits.set(0, size);
            } else {
              int firstSetBit = bits.nextSetBit(0);
              int bitToFlip = bits.nextClearBit(firstSetBit);

              if (bitToFlip == index.size()) {
                return endOfData();
              }

              /*
               * The current set in sorted order looks like
               * {firstSetBit, firstSetBit + 1, ..., bitToFlip - 1, ...}
               * where it does *not* contain bitToFlip.
               *
               * The next combination is
               *
               * {0, 1, ..., bitToFlip - firstSetBit - 2, bitToFlip, ...}
               *
               * This is lexicographically next if you look at the combinations in descending order
               * e.g. {2, 1, 0}, {3, 1, 0}, {3, 2, 0}, {3, 2, 1}, {4, 1, 0}...
               */

              bits.set(0, bitToFlip - firstSetBit - 1);
              bits.clear(bitToFlip - firstSetBit - 1, bitToFlip);
              bits.set(bitToFlip);
            }
            final BitSet copy = (BitSet) bits.clone();
            return new AbstractSet<E>() {
              @Override
              public boolean contains(@Nullable Object o) {
                Integer i = index.get(o);
                return i != null && copy.get(i);
              }

              @Override
              public Iterator<E> iterator() {
                return new AbstractIterator<E>() {
                  int i = -1;

                  @Override
                  protected E computeNext() {
                    i = copy.nextSetBit(i + 1);
                    if (i == -1) {
                      return endOfData();
                    }
                    return index.keySet().asList().get(i);
                  }
                };
              }

              @Override
              public int size() {
                return size;
              }
            };
          }
        };
      }

      @Override
      public int size() {
        return IntMath.binomial(index.size(), size);
      }

      @Override
      public String toString() {
        return "Sets.combinations(" + index.keySet() + ", " + size + ")";
      }
    };
  }

  
An implementation for Set.hashCode(). /** An implementation for {@link Set#hashCode()}. */
  static int hashCodeImpl(Set<?> s) {
    int hashCode = 0;
    for (Object o : s) {
      hashCode += o != null ? o.hashCode() : 0;

      hashCode = ~~hashCode;
      // Needed to deal with unusual integer overflow in GWT.
    }
    return hashCode;
  }

  
An implementation for Set.equals(Object). /** An implementation for {@link Set#equals(Object)}. */
  static boolean equalsImpl(Set<?> s, @Nullable Object object) {
    if (s == object) {
      return true;
    }
    if (object instanceof Set) {
      Set<?> o = (Set<?>) object;

      try {
        return s.size() == o.size() && s.containsAll(o);
      } catch (NullPointerException | ClassCastException ignored) {
        return false;
      }
    }
    return false;
  }

  
Returns an unmodifiable view of the specified navigable set. This method allows modules to provide users with "read-only" access to internal navigable sets. Query operations on the returned set "read through" to the specified set, and attempts to modify the returned set, whether direct or via its collection views, result in an UnsupportedOperationException. 
The returned navigable set will be serializable if the specified navigable set is
serializable.
Params:
set – the navigable set for which an unmodifiable view is to be returned
Returns:
an unmodifiable view of the specified navigable set
Since:
12.0/**
   * Returns an unmodifiable view of the specified navigable set. This method allows modules to
   * provide users with "read-only" access to internal navigable sets. Query operations on the
   * returned set "read through" to the specified set, and attempts to modify the returned set,
   * whether direct or via its collection views, result in an {@code UnsupportedOperationException}.
   *
   * <p>The returned navigable set will be serializable if the specified navigable set is
   * serializable.
   *
   * @param set the navigable set for which an unmodifiable view is to be returned
   * @return an unmodifiable view of the specified navigable set
   * @since 12.0
   */
  public static <E> NavigableSet<E> unmodifiableNavigableSet(NavigableSet<E> set) {
    if (set instanceof ImmutableCollection || set instanceof UnmodifiableNavigableSet) {
      return set;
    }
    return new UnmodifiableNavigableSet<E>(set);
  }

  static final class UnmodifiableNavigableSet<E> extends ForwardingSortedSet<E>
      implements NavigableSet<E>, Serializable {
    private final NavigableSet<E> delegate;
    private final SortedSet<E> unmodifiableDelegate;

    UnmodifiableNavigableSet(NavigableSet<E> delegate) {
      this.delegate = checkNotNull(delegate);
      this.unmodifiableDelegate = Collections.unmodifiableSortedSet(delegate);
    }

    @Override
    protected SortedSet<E> delegate() {
      return unmodifiableDelegate;
    }

    // default methods not forwarded by ForwardingSortedSet

    @Override
    public boolean removeIf(java.util.function.Predicate<? super E> filter) {
      throw new UnsupportedOperationException();
    }

    @Override
    public Stream<E> stream() {
      return delegate.stream();
    }

    @Override
    public Stream<E> parallelStream() {
      return delegate.parallelStream();
    }

    @Override
    public void forEach(Consumer<? super E> action) {
      delegate.forEach(action);
    }

    @Override
    public E lower(E e) {
      return delegate.lower(e);
    }

    @Override
    public E floor(E e) {
      return delegate.floor(e);
    }

    @Override
    public E ceiling(E e) {
      return delegate.ceiling(e);
    }

    @Override
    public E higher(E e) {
      return delegate.higher(e);
    }

    @Override
    public E pollFirst() {
      throw new UnsupportedOperationException();
    }

    @Override
    public E pollLast() {
      throw new UnsupportedOperationException();
    }

    private transient @MonotonicNonNull UnmodifiableNavigableSet<E> descendingSet;

    @Override
    public NavigableSet<E> descendingSet() {
      UnmodifiableNavigableSet<E> result = descendingSet;
      if (result == null) {
        result = descendingSet = new UnmodifiableNavigableSet<E>(delegate.descendingSet());
        result.descendingSet = this;
      }
      return result;
    }

    @Override
    public Iterator<E> descendingIterator() {
      return Iterators.unmodifiableIterator(delegate.descendingIterator());
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return unmodifiableNavigableSet(
          delegate.subSet(fromElement, fromInclusive, toElement, toInclusive));
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return unmodifiableNavigableSet(delegate.headSet(toElement, inclusive));
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return unmodifiableNavigableSet(delegate.tailSet(fromElement, inclusive));
    }

    private static final long serialVersionUID = 0;
  }

  
Returns a synchronized (thread-safe) navigable set backed by the specified navigable set. In
order to guarantee serial access, it is critical that all access to the backing
navigable set is accomplished through the returned navigable set (or its views).
It is imperative that the user manually synchronize on the returned sorted set when iterating over it or any of its descendingSet, subSet, headSet, or tailSet views. 

NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
 ...
synchronized (set) {
  // Must be in the synchronized block
  Iterator<E> it = set.iterator();
  while (it.hasNext()) {
    foo(it.next());
  }
 }

or:

NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
NavigableSet<E> set2 = set.descendingSet().headSet(foo);
 ...
synchronized (set) { // Note: set, not set2!!!
  // Must be in the synchronized block
  Iterator<E> it = set2.descendingIterator();
  while (it.hasNext())
    foo(it.next());
  }
 }
Failure to follow this advice may result in non-deterministic behavior.
The returned navigable set will be serializable if the specified navigable set is
serializable.
Params:
navigableSet – the navigable set to be "wrapped" in a synchronized navigable set.
Returns:
a synchronized view of the specified navigable set.
Since:
13.0/**
   * Returns a synchronized (thread-safe) navigable set backed by the specified navigable set. In
   * order to guarantee serial access, it is critical that <b>all</b> access to the backing
   * navigable set is accomplished through the returned navigable set (or its views).
   *
   * <p>It is imperative that the user manually synchronize on the returned sorted set when
   * iterating over it or any of its {@code descendingSet}, {@code subSet}, {@code headSet}, or
   * {@code tailSet} views.
   *
   * <pre>{@code
   * NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
   *  ...
   * synchronized (set) {
   *   // Must be in the synchronized block
   *   Iterator<E> it = set.iterator();
   *   while (it.hasNext()) {
   *     foo(it.next());
   *   }
   * }
   * }</pre>
   *
   * <p>or:
   *
   * <pre>{@code
   * NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
   * NavigableSet<E> set2 = set.descendingSet().headSet(foo);
   *  ...
   * synchronized (set) { // Note: set, not set2!!!
   *   // Must be in the synchronized block
   *   Iterator<E> it = set2.descendingIterator();
   *   while (it.hasNext())
   *     foo(it.next());
   *   }
   * }
   * }</pre>
   *
   * <p>Failure to follow this advice may result in non-deterministic behavior.
   *
   * <p>The returned navigable set will be serializable if the specified navigable set is
   * serializable.
   *
   * @param navigableSet the navigable set to be "wrapped" in a synchronized navigable set.
   * @return a synchronized view of the specified navigable set.
   * @since 13.0
   */
  @GwtIncompatible // NavigableSet
  public static <E> NavigableSet<E> synchronizedNavigableSet(NavigableSet<E> navigableSet) {
    return Synchronized.navigableSet(navigableSet);
  }

  
Remove each element in an iterable from a set. /** Remove each element in an iterable from a set. */
  static boolean removeAllImpl(Set<?> set, Iterator<?> iterator) {
    boolean changed = false;
    while (iterator.hasNext()) {
      changed |= set.remove(iterator.next());
    }
    return changed;
  }

  static boolean removeAllImpl(Set<?> set, Collection<?> collection) {
    checkNotNull(collection); // for GWT
    if (collection instanceof Multiset) {
      collection = ((Multiset<?>) collection).elementSet();
    }
    /*
     * AbstractSet.removeAll(List) has quadratic behavior if the list size
     * is just more than the set's size.  We augment the test by
     * assuming that sets have fast contains() performance, and other
     * collections don't.  See
     * http://code.google.com/p/guava-libraries/issues/detail?id=1013
     */
    if (collection instanceof Set && collection.size() > set.size()) {
      return Iterators.removeAll(set.iterator(), collection);
    } else {
      return removeAllImpl(set, collection.iterator());
    }
  }

  @GwtIncompatible // NavigableSet
  static class DescendingSet<E> extends ForwardingNavigableSet<E> {
    private final NavigableSet<E> forward;

    DescendingSet(NavigableSet<E> forward) {
      this.forward = forward;
    }

    @Override
    protected NavigableSet<E> delegate() {
      return forward;
    }

    @Override
    public E lower(E e) {
      return forward.higher(e);
    }

    @Override
    public E floor(E e) {
      return forward.ceiling(e);
    }

    @Override
    public E ceiling(E e) {
      return forward.floor(e);
    }

    @Override
    public E higher(E e) {
      return forward.lower(e);
    }

    @Override
    public E pollFirst() {
      return forward.pollLast();
    }

    @Override
    public E pollLast() {
      return forward.pollFirst();
    }

    @Override
    public NavigableSet<E> descendingSet() {
      return forward;
    }

    @Override
    public Iterator<E> descendingIterator() {
      return forward.iterator();
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return forward.subSet(toElement, toInclusive, fromElement, fromInclusive).descendingSet();
    }

    @Override
    public SortedSet<E> subSet(E fromElement, E toElement) {
      return standardSubSet(fromElement, toElement);
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return forward.tailSet(toElement, inclusive).descendingSet();
    }

    @Override
    public SortedSet<E> headSet(E toElement) {
      return standardHeadSet(toElement);
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return forward.headSet(fromElement, inclusive).descendingSet();
    }

    @Override
    public SortedSet<E> tailSet(E fromElement) {
      return standardTailSet(fromElement);
    }

    @SuppressWarnings("unchecked")
    @Override
    public Comparator<? super E> comparator() {
      Comparator<? super E> forwardComparator = forward.comparator();
      if (forwardComparator == null) {
        return (Comparator) Ordering.natural().reverse();
      } else {
        return reverse(forwardComparator);
      }
    }

    // If we inline this, we get a javac error.
    private static <T> Ordering<T> reverse(Comparator<T> forward) {
      return Ordering.from(forward).reverse();
    }

    @Override
    public E first() {
      return forward.last();
    }

    @Override
    public E last() {
      return forward.first();
    }

    @Override
    public Iterator<E> iterator() {
      return forward.descendingIterator();
    }

    @Override
    public Object[] toArray() {
      return standardToArray();
    }

    @Override
    public <T> T[] toArray(T[] array) {
      return standardToArray(array);
    }

    @Override
    public String toString() {
      return standardToString();
    }
  }

  
Returns a view of the portion of set whose elements are contained by range. 
This method delegates to the appropriate methods of NavigableSet (namely subSet(), tailSet(), and headSet()) to actually construct the view. Consult these methods for a full description of the returned view's behavior. 
Warning: Ranges always represent a range of values using the values' natural ordering. NavigableSet on the other hand can specify a custom ordering via a Comparator, which can violate the natural ordering. Using this method (or in general using Range) with unnaturally-ordered sets can lead to unexpected and undefined behavior. 
Since:
20.0/**
   * Returns a view of the portion of {@code set} whose elements are contained by {@code range}.
   *
   * <p>This method delegates to the appropriate methods of {@link NavigableSet} (namely {@link
   * NavigableSet#subSet(Object, boolean, Object, boolean) subSet()}, {@link
   * NavigableSet#tailSet(Object, boolean) tailSet()}, and {@link NavigableSet#headSet(Object,
   * boolean) headSet()}) to actually construct the view. Consult these methods for a full
   * description of the returned view's behavior.
   *
   * <p><b>Warning:</b> {@code Range}s always represent a range of values using the values' natural
   * ordering. {@code NavigableSet} on the other hand can specify a custom ordering via a {@link
   * Comparator}, which can violate the natural ordering. Using this method (or in general using
   * {@code Range}) with unnaturally-ordered sets can lead to unexpected and undefined behavior.
   *
   * @since 20.0
   */
  @Beta
  @GwtIncompatible // NavigableSet
  public static <K extends Comparable<? super K>> NavigableSet<K> subSet(
      NavigableSet<K> set, Range<K> range) {
    if (set.comparator() != null
        && set.comparator() != Ordering.natural()
        && range.hasLowerBound()
        && range.hasUpperBound()) {
      checkArgument(
          set.comparator().compare(range.lowerEndpoint(), range.upperEndpoint()) <= 0,
          "set is using a custom comparator which is inconsistent with the natural ordering.");
    }
    if (range.hasLowerBound() && range.hasUpperBound()) {
      return set.subSet(
          range.lowerEndpoint(),
          range.lowerBoundType() == BoundType.CLOSED,
          range.upperEndpoint(),
          range.upperBoundType() == BoundType.CLOSED);
    } else if (range.hasLowerBound()) {
      return set.tailSet(range.lowerEndpoint(), range.lowerBoundType() == BoundType.CLOSED);
    } else if (range.hasUpperBound()) {
      return set.headSet(range.upperEndpoint(), range.upperBoundType() == BoundType.CLOSED);
    }
    return checkNotNull(set);
  }
}