# SetAlgebra A type that provides mathematical set operations. ``` protocol SetAlgebra : Equatable, ExpressibleByArrayLiteral ``` ## Overview 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)` implies `x.union(y) == y` - `x.isSuperset(of: y)` implies `x.union(y) == x` - `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` ## Topics ### Creating a Set [`init()`](/documentation/Swift/SetAlgebra/init()) Creates an empty set. ### Testing for Membership [`contains(_:)`](/documentation/Swift/SetAlgebra/contains(_:)) Returns a Boolean value that indicates whether the given element exists in the set. [`Element`](/documentation/Swift/SetAlgebra/Element) A type for which the conforming type provides a containment test. ### Adding and Removing Elements [`insert(_:)`](/documentation/Swift/SetAlgebra/insert(_:)) Inserts the given element in the set if it is not already present. [`update(with:)`](/documentation/Swift/SetAlgebra/update(with:)) Inserts the given element into the set unconditionally. [`remove(_:)`](/documentation/Swift/SetAlgebra/remove(_:)) Removes the given element and any elements subsumed by the given element. ### Combining Sets [`union(_:)`](/documentation/Swift/SetAlgebra/union(_:)) Returns a new set with the elements of both this and the given set. [`formUnion(_:)`](/documentation/Swift/SetAlgebra/formUnion(_:)) Adds the elements of the given set to the set. [`intersection(_:)`](/documentation/Swift/SetAlgebra/intersection(_:)) Returns a new set with the elements that are common to both this set and the given set. [`formIntersection(_:)`](/documentation/Swift/SetAlgebra/formIntersection(_:)) Removes the elements of this set that aren’t also in the given set. [`symmetricDifference(_:)`](/documentation/Swift/SetAlgebra/symmetricDifference(_:)) Returns a new set with the elements that are either in this set or in the given set, but not in both. [`formSymmetricDifference(_:)`](/documentation/Swift/SetAlgebra/formSymmetricDifference(_:)) 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. ### Comparing Sets [`isStrictSubset(of:)`](/documentation/Swift/SetAlgebra/isStrictSubset(of:)) Returns a Boolean value that indicates whether this set is a strict subset of the given set. [`isStrictSuperset(of:)`](/documentation/Swift/SetAlgebra/isStrictSuperset(of:)) Returns a Boolean value that indicates whether this set is a strict superset of the given set. --- Copyright © 2026 Apple Inc. All rights reserved. | [Terms of Use](https://www.apple.com/legal/internet-services/terms/site.html) | [Privacy Policy](https://www.apple.com/privacy/privacy-policy)