## SetAlgebra

`protocol SetAlgebra`

A type that provides mathematical set operations.

You use types that conform to the `SetAlgebra` protocol when you need efficient membership tests or mathematical set operations such as intersection, union, and subtraction. In the standard library, you can use the `Set` type with elements of any hashable type, or you can easily create bit masks with `SetAlgebra` conformance using the `OptionSet` protocol. See those types for more information.

Note: Unlike ordinary set types, the `Element` type of an `OptionSet` is identical to the `OptionSet` type itself. The `SetAlgebra` protocol is specifically designed to accommodate both kinds of set.

# Conforming to the SetAlgebra Protocol

When implementing a custom type that conforms to the `SetAlgebra` protocol, you must implement the required initializers and methods. For the inherited methods to work properly, conforming types must meet the following axioms. Assume that `S` is a custom type that conforms to the `SetAlgebra` protocol, `x` and `y` are instances of `S`, and `e` is of type `S.Element`---the type that the set holds.

• `S() == []`
• `x.intersection(x) == x`
• `x.intersection([]) == []`
• `x.union(x) == x`
• `x.union([]) == x`
• `x.contains(e)` implies `x.union(y).contains(e)`
• `x.union(y).contains(e)` implies `x.contains(e) || y.contains(e)`
• `x.contains(e) && y.contains(e)` if and only if `x.intersection(y).contains(e)` `x.isSubset(of:** y)` if and only if `y.isSuperset(of: x)` `x.isStrictSuperset(of:** y)` if and only if `x.isSuperset(of: y) && x != y` **`x.isStrictSubset(of:** y)` if and only if `x.isSubset(of: y) && x != y`

See Also: `OptionSet`, `Set`

Inheritance `Equatable, ExpressibleByArrayLiteral` View Protocol Hierarchy → `Element` A type for which the conforming type provides a containment test. `import Swift`

### Initializers

init() Required

Creates an empty set.

This initializer is equivalent to initializing with an empty array literal. For example, you create an empty `Set` instance with either this initializer or with an empty array literal.

``````var emptySet = Set<Int>()
print(emptySet.isEmpty)
// Prints "true"

emptySet = []
print(emptySet.isEmpty)
// Prints "true"``````

#### Declaration

`init()`
init(_:)

Creates a new set from a finite sequence of items.

Use this initializer to create a new set from an existing sequence, like an array or a range:

``````let validIndices = Set(0..<7).subtracting([2, 4, 5])
print(validIndices)
// Prints "[6, 0, 1, 3]"``````

`sequence`: The elements to use as members of the new set.

#### Declaration

`init<S : Sequence where S.Iterator.Element == Element>(_ sequence: S)`
init(arrayLiteral:)

Creates an instance initialized with the given elements.

#### Declaration

`init(arrayLiteral elements: Self.Element...)`

#### Declared In

`ExpressibleByArrayLiteral`

### Instance Variables

var isEmpty: Bool

A Boolean value that indicates whether the set has no elements.

#### Declaration

`var isEmpty: Bool { get }`

### Instance Methods

func ==(_:rhs:) Required

Returns a Boolean value indicating whether two values are equal.

Equality is the inverse of inequality. For any values `a` and `b`, `a == b` implies that `a != b` is `false`.

Parameters: lhs: A value to compare. rhs: Another value to compare.

#### Declaration

`func ==(lhs: Self, rhs: Self) -> Bool`

#### Declared In

`Equatable`
func contains(_:) Required

Returns a Boolean value that indicates whether the given element exists in the set.

This example uses the `contains(_:)` method to test whether an integer is a member of a set of prime numbers.

``````let primes: Set = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
let x = 5
if primes.contains(x) {
print("\(x) is prime!")
} else {
print("\(x). Not prime.")
}
// Prints "5 is prime!"``````

`member`: An element to look for in the set. Returns: `true` if `member` exists in the set; otherwise, `false`.

#### Declaration

`func contains(_ member: Self.Element) -> Bool`
mutating func formIntersection(_:) Required

Removes the elements of this set that aren't also in the given set.

In the following example, the elements of the `employees` set that are not also members of the `neighbors` set are removed. In particular, the names `"Alicia"`, `"Chris"`, and `"Diana"` are removed.

``````var employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
employees.formIntersection(neighbors)
print(employees)
// Prints "["Bethany", "Eric"]"``````

`other`: A set of the same type as the current set.

#### Declaration

`mutating func formIntersection(_ other: Self)`
mutating func formSymmetricDifference(_:) Required

Removes the elements of the set that are also in the given set and adds the members of the given set that are not already in the set.

In the following example, the elements of the `employees` set that are also members of `neighbors` are removed from `employees`, while the elements of `neighbors` that are not members of `employees` are added to `employees`. In particular, the names `"Alicia"`, `"Chris"`, and `"Diana"` are removed from `employees` while the names `"Forlani"` and `"Greta"` are added.

``````var employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
employees.formSymmetricDifference(neighbors)
print(employees)
// Prints "["Diana", "Chris", "Forlani", "Alicia", "Greta"]"``````

`other`: A set of the same type.

#### Declaration

`mutating func formSymmetricDifference(_ other: Self)`
mutating func formUnion(_:) Required

Adds the elements of the given set to the set.

In the following example, the elements of the `visitors` set are added to the `attendees` set:

``````var attendees: Set = ["Alicia", "Bethany", "Diana"]
let visitors: Set = ["Marcia", "Nathaniel"]
attendees.formUnion(visitors)
print(attendees)
// Prints "["Diana", "Nathaniel", "Bethany", "Alicia", "Marcia"]"``````

If the set already contains one or more elements that are also in `other`, the existing members are kept.

``````var initialIndices = Set(0..<5)
initialIndices.formUnion([2, 3, 6, 7])
print(initialIndices)
// Prints "[2, 4, 6, 7, 0, 1, 3]"``````

`other`: A set of the same type as the current set.

#### Declaration

`mutating func formUnion(_ other: Self)`
mutating func insert(_:) Required

Inserts the given element in the set if it is not already present.

If an element equal to `newMember` is already contained in the set, this method has no effect. In this example, a new element is inserted into `classDays`, a set of days of the week. When an existing element is inserted, the `classDays` set does not change.

``````enum DayOfTheWeek: Int {
case sunday, monday, tuesday, wednesday, thursday,
friday, saturday
}

var classDays: Set<DayOfTheWeek> = [.wednesday, .friday]
print(classDays.insert(.monday))
// Prints "(true, .monday)"
print(classDays)
// Prints "[.friday, .wednesday, .monday]"

print(classDays.insert(.friday))
// Prints "(false, .friday)"
print(classDays)
// Prints "[.friday, .wednesday, .monday]"``````

`newMember`: An element to insert into the set. Returns: `(true, newMember)` if `newMember` was not contained in the set. If an element equal to `newMember` was already contained in the set, the method returns `(false, oldMember)`, where `oldMember` is the element that was equal to `newMember`. In some cases, `oldMember` may be distinguishable from `newMember` by identity comparison or some other means.

#### Declaration

`mutating func insert(_ newMember: Self.Element) -> (inserted: Bool, memberAfterInsert: Self.Element)`
func intersection(_:) Required

Returns a new set with the elements that are common to both this set and the given set.

In the following example, the `bothNeighborsAndEmployees` set is made up of the elements that are in both the `employees` and `neighbors` sets. Elements that are in either one or the other, but not both, are left out of the result of the intersection.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
let bothNeighborsAndEmployees = employees.intersection(neighbors)
print(bothNeighborsAndEmployees)
// Prints "["Bethany", "Eric"]"``````

`other`: A set of the same type as the current set. Returns: A new set.

Note: if this set and `other` contain elements that are equal but distinguishable (e.g. via `===`), which of these elements is present in the result is unspecified.

#### Declaration

`func intersection(_ other: Self) -> Self`
func isDisjoint(with:)

Returns a Boolean value that indicates whether the set has no members in common with the given set.

In the following example, the `employees` set is disjoint with the `visitors` set because no name appears in both sets.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let visitors: Set = ["Marcia", "Nathaniel", "Olivia"]
print(employees.isDisjoint(with: visitors))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set has no elements in common with `other`; otherwise, `false`.

#### Declaration

`func isDisjoint(with other: Self) -> Bool`
func isSubset(of:)

Returns a Boolean value that indicates whether the set is a subset of another set.

Set A is a subset of another set B if every member of A is also a member of B.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(attendees.isSubset(of: employees))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a subset of `other`; otherwise, `false`.

#### Declaration

`func isSubset(of other: Self) -> Bool`
func isSuperset(of:)

Returns a Boolean value that indicates whether the set is a superset of the given set.

Set A is a superset of another set B if every member of B is also a member of A.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(employees.isSuperset(of: attendees))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a superset of `possibleSubset`; otherwise, `false`.

#### Declaration

`func isSuperset(of other: Self) -> Bool`
mutating func remove(_:) Required

Removes the given element and any elements subsumed by the given element.

`member`: The element of the set to remove. Returns: For ordinary sets, an element equal to `member` if `member` is contained in the set; otherwise, `nil`. In some cases, a returned element may be distinguishable from `newMember` by identity comparison or some other means.

For sets where the set type and element type are the same, like `OptionSet` types, this method returns any intersection between the set and `[member]`, or `nil` if the intersection is empty.

#### Declaration

`mutating func remove(_ member: Self.Element) -> Self.Element?`
mutating func subtract(_:)

Removes the elements of the given set from this set.

In the following example, the elements of the `employees` set that are also members of the `neighbors` set are removed. In particular, the names `"Bethany"` and `"Eric"` are removed from `employees`.

``````var employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
employees.subtract(neighbors)
print(employees)
// Prints "["Diana", "Chris", "Alicia"]"``````

`other`: A set of the same type as the current set.

#### Declaration

`mutating func subtract(_ other: Self)`
func subtracting(_:)

Returns a new set containing the elements of this set that do not occur in the given set.

In the following example, the `nonNeighbors` set is made up of the elements of the `employees` set that are not elements of `neighbors`:

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
let nonNeighbors = employees.subtracting(neighbors)
print(nonNeighbors)
// Prints "["Diana", "Chris", "Alicia"]"``````

`other`: A set of the same type as the current set. Returns: A new set.

#### Declaration

`func subtracting(_ other: Self) -> Self`
func symmetricDifference(_:) Required

Returns a new set with the elements that are either in this set or in the given set, but not in both.

In the following example, the `eitherNeighborsOrEmployees` set is made up of the elements of the `employees` and `neighbors` sets that are not in both `employees` and `neighbors`. In particular, the names `"Bethany"` and `"Eric"` do not appear in `eitherNeighborsOrEmployees`.

``````let employees: Set = ["Alicia", "Bethany", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani"]
let eitherNeighborsOrEmployees = employees.symmetricDifference(neighbors)
print(eitherNeighborsOrEmployees)
// Prints "["Diana", "Forlani", "Alicia"]"``````

`other`: A set of the same type as the current set. Returns: A new set.

#### Declaration

`func symmetricDifference(_ other: Self) -> Self`
func union(_:) Required

Returns a new set with the elements of both this and the given set.

In the following example, the `attendeesAndVisitors` set is made up of the elements of the `attendees` and `visitors` sets:

``````let attendees: Set = ["Alicia", "Bethany", "Diana"]
let visitors = ["Marcia", "Nathaniel"]
let attendeesAndVisitors = attendees.union(visitors)
print(attendeesAndVisitors)
// Prints "["Diana", "Nathaniel", "Bethany", "Alicia", "Marcia"]"``````

If the set already contains one or more elements that are also in `other`, the existing members are kept.

``````let initialIndices = Set(0..<5)
let expandedIndices = initialIndices.union([2, 3, 6, 7])
print(expandedIndices)
// Prints "[2, 4, 6, 7, 0, 1, 3]"``````

`other`: A set of the same type as the current set. Returns: A new set with the unique elements of this set and `other`.

Note: if this set and `other` contain elements that are equal but distinguishable (e.g. via `===`), which of these elements is present in the result is unspecified.

#### Declaration

`func union(_ other: Self) -> Self`
mutating func update(with:) Required

Inserts the given element into the set unconditionally.

If an element equal to `newMember` is already contained in the set, `newMember` replaces the existing element. In this example, an existing element is inserted into `classDays`, a set of days of the week.

``````enum DayOfTheWeek: Int {
case sunday, monday, tuesday, wednesday, thursday,
friday, saturday
}

var classDays: Set<DayOfTheWeek> = [.monday, .wednesday, .friday]
print(classDays.update(with: .monday))
// Prints "Optional(.monday)"``````

`newMember`: An element to insert into the set. Returns: For ordinary sets, an element equal to `newMember` if the set already contained such a member; otherwise, `nil`. In some cases, the returned element may be distinguishable from `newMember` by identity comparison or some other means.

For sets where the set type and element type are the same, like `OptionSet` types, this method returns any intersection between the set and `[newMember]`, or `nil` if the intersection is empty.

#### Declaration

`mutating func update(with newMember: Self.Element) -> Self.Element?`

### Default Implementations

init(_:)

Creates a new set from a finite sequence of items.

Use this initializer to create a new set from an existing sequence, like an array or a range:

``````let validIndices = Set(0..<7).subtracting([2, 4, 5])
print(validIndices)
// Prints "[6, 0, 1, 3]"``````

`sequence`: The elements to use as members of the new set.

#### Declaration

`init<S : Sequence where S.Iterator.Element == Element>(_ sequence: S)`
init(arrayLiteral:)

Creates a set containing the elements of the given array literal.

Do not call this initializer directly. It is used by the compiler when you use an array literal. Instead, create a new set using an array literal as its value by enclosing a comma-separated list of values in square brackets. You can use an array literal anywhere a set is expected by the type context.

Here, a set of strings is created from an array literal holding only strings:

``````let ingredients: Set = ["cocoa beans", "sugar", "cocoa butter", "salt"]
if ingredients.isSuperset(of: ["sugar", "salt"]) {
print("Whatever it is, it's bound to be delicious!")
}
// Prints "Whatever it is, it's bound to be delicious!"``````

`arrayLiteral`: A list of elements of the new set.

#### Declaration

`init(arrayLiteral: Self.Element...)`
var isEmpty: Bool

A Boolean value that indicates whether the set has no elements.

#### Declaration

`var isEmpty: Bool { get }`
func isDisjoint(with:)

Returns a Boolean value that indicates whether the set has no members in common with the given set.

In the following example, the `employees` set is disjoint with the `visitors` set because no name appears in both sets.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let visitors: Set = ["Marcia", "Nathaniel", "Olivia"]
print(employees.isDisjoint(with: visitors))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set has no elements in common with `other`; otherwise, `false`.

#### Declaration

`func isDisjoint(with other: Self) -> Bool`
func isStrictSubset(of:)

Returns a Boolean value that indicates whether this set is a strict subset of the given set.

Set A is a strict subset of another set B if every member of A is also a member of B and B contains at least one element that is not a member of A.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(attendees.isStrictSubset(of: employees))
// Prints "true"

// A set is never a strict subset of itself:
print(attendees.isStrictSubset(of: attendees))
// Prints "false"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a strict subset of `other`; otherwise, `false`.

#### Declaration

`func isStrictSubset(of other: Self) -> Bool`
func isStrictSuperset(of:)

Returns a Boolean value that indicates whether this set is a strict superset of the given set.

Set A is a strict superset of another set B if every member of B is also a member of A and A contains at least one element that is not a member of B.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(employees.isStrictSuperset(of: attendees))
// Prints "true"

// A set is never a strict superset of itself:
print(employees.isStrictSuperset(of: employees))
// Prints "false"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a strict superset of `other`; otherwise, `false`.

#### Declaration

`func isStrictSuperset(of other: Self) -> Bool`
func isSubset(of:)

Returns a Boolean value that indicates whether the set is a subset of another set.

Set A is a subset of another set B if every member of A is also a member of B.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(attendees.isSubset(of: employees))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a subset of `other`; otherwise, `false`.

#### Declaration

`func isSubset(of other: Self) -> Bool`
func isSuperset(of:)

Returns a Boolean value that indicates whether the set is a superset of the given set.

Set A is a superset of another set B if every member of B is also a member of A.

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let attendees: Set = ["Alicia", "Bethany", "Diana"]
print(employees.isSuperset(of: attendees))
// Prints "true"``````

`other`: A set of the same type as the current set. Returns: `true` if the set is a superset of `other`; otherwise, `false`.

#### Declaration

`func isSuperset(of other: Self) -> Bool`
mutating func subtract(_:)

Removes the elements of the given set from this set.

In the following example, the elements of the `employees` set that are also members of the `neighbors` set are removed. In particular, the names `"Bethany"` and `"Eric"` are removed from `employees`.

``````var employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
employees.subtract(neighbors)
print(employees)
// Prints "["Diana", "Chris", "Alicia"]"``````

`other`: A set of the same type as the current set.

#### Declaration

`mutating func subtract(_ other: Self)`
func subtracting(_:)

Returns a new set containing the elements of this set that do not occur in the given set.

In the following example, the `nonNeighbors` set is made up of the elements of the `employees` set that are not elements of `neighbors`:

``````let employees: Set = ["Alicia", "Bethany", "Chris", "Diana", "Eric"]
let neighbors: Set = ["Bethany", "Eric", "Forlani", "Greta"]
let nonNeighbors = employees.subtract(neighbors)
print(nonNeighbors)
// Prints "["Diana", "Chris", "Alicia"]"``````

`other`: A set of the same type as the current set. Returns: A new set.

#### Declaration

`func subtracting(_ other: Self) -> Self`