---
title: "Operations on Sets"
language: "en"
type: "Guide"
summary: "In the Wolfram Language, sets are represented by sorted lists."
sections: 
  - 
    title: "Testing Relations between Sets"
    link: "null"
  - 
    title: "Testing Set Containment"
    link: "null"
  - 
    title: "Testing Elements in Sets"
    link: "null"
  - 
    title: "Operations on Subsets"
    link: "null"
  - 
    title: "Constructing Sets from Lists"
    link: "null"
  - 
    title: "Symbolic Set Theory"
    link: "null"
keywords: 
- set
- mathematical set
- set operation
- set theory
- set algebra
- Boolean algebra
- set theoretic operation
- set testing
- membership testing
- member
- set membership
- element
- Boolean operation
- Boolean simplification
- mathematical logic
- binary operations on sets
- binary operation
- binary relation
- set arithmetic
- union
- complement
- inersection
- disjoint sets
- inclusion
- containment
- sorted list
- ordered list
- set element
- conjunction
- disjunction
- truth value
canonical_url: "https://reference.wolfram.com/language/guide/OperationsOnSets.html"
source: "Wolfram Language Documentation"
related_guides: 
  - 
    title: "Logic and Boolean Algebra"
    link: "https://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.en.md"
  - 
    title: "List Manipulation"
    link: "https://reference.wolfram.com/language/guide/ListManipulation.en.md"
related_functions: 
  - 
    title: "Complement"
    link: "https://reference.wolfram.com/language/ref/Complement.en.md"
  - 
    title: "ContainsAll"
    link: "https://reference.wolfram.com/language/ref/ContainsAll.en.md"
  - 
    title: "ContainsAny"
    link: "https://reference.wolfram.com/language/ref/ContainsAny.en.md"
  - 
    title: "ContainsExactly"
    link: "https://reference.wolfram.com/language/ref/ContainsExactly.en.md"
  - 
    title: "ContainsNone"
    link: "https://reference.wolfram.com/language/ref/ContainsNone.en.md"
  - 
    title: "ContainsOnly"
    link: "https://reference.wolfram.com/language/ref/ContainsOnly.en.md"
  - 
    title: "DeleteAdjacentDuplicates"
    link: "https://reference.wolfram.com/language/ref/DeleteAdjacentDuplicates.en.md"
  - 
    title: "DeleteDuplicates"
    link: "https://reference.wolfram.com/language/ref/DeleteDuplicates.en.md"
  - 
    title: "DisjointQ"
    link: "https://reference.wolfram.com/language/ref/DisjointQ.en.md"
  - 
    title: "DuplicateFreeQ"
    link: "https://reference.wolfram.com/language/ref/DuplicateFreeQ.en.md"
  - 
    title: "Element"
    link: "https://reference.wolfram.com/language/ref/Element.en.md"
  - 
    title: "Exists"
    link: "https://reference.wolfram.com/language/ref/Exists.en.md"
  - 
    title: "ForAll"
    link: "https://reference.wolfram.com/language/ref/ForAll.en.md"
  - 
    title: "FreeQ"
    link: "https://reference.wolfram.com/language/ref/FreeQ.en.md"
  - 
    title: "Groupings"
    link: "https://reference.wolfram.com/language/ref/Groupings.en.md"
  - 
    title: "IntersectingQ"
    link: "https://reference.wolfram.com/language/ref/IntersectingQ.en.md"
  - 
    title: "Intersection"
    link: "https://reference.wolfram.com/language/ref/Intersection.en.md"
  - 
    title: "MemberQ"
    link: "https://reference.wolfram.com/language/ref/MemberQ.en.md"
  - 
    title: "Resolve"
    link: "https://reference.wolfram.com/language/ref/Resolve.en.md"
  - 
    title: "Sort"
    link: "https://reference.wolfram.com/language/ref/Sort.en.md"
  - 
    title: "SubsetCases"
    link: "https://reference.wolfram.com/language/ref/SubsetCases.en.md"
  - 
    title: "SubsetCount"
    link: "https://reference.wolfram.com/language/ref/SubsetCount.en.md"
  - 
    title: "SubsetPosition"
    link: "https://reference.wolfram.com/language/ref/SubsetPosition.en.md"
  - 
    title: "SubsetQ"
    link: "https://reference.wolfram.com/language/ref/SubsetQ.en.md"
  - 
    title: "SubsetReplace"
    link: "https://reference.wolfram.com/language/ref/SubsetReplace.en.md"
  - 
    title: "Subsets"
    link: "https://reference.wolfram.com/language/ref/Subsets.en.md"
  - 
    title: "SymmetricDifference"
    link: "https://reference.wolfram.com/language/ref/SymmetricDifference.en.md"
  - 
    title: "Tuples"
    link: "https://reference.wolfram.com/language/ref/Tuples.en.md"
  - 
    title: "Union"
    link: "https://reference.wolfram.com/language/ref/Union.en.md"
  - 
    title: "UniqueElements"
    link: "https://reference.wolfram.com/language/ref/UniqueElements.en.md"
---
# Operations on Sets

In the Wolfram Language, sets are represented by sorted lists.

---

[`Union`](https://reference.wolfram.com/language/ref/Union.en.md) — find the union of any number of sets

[`Intersection`](https://reference.wolfram.com/language/ref/Intersection.en.md) — find the intersection of any number of sets

[`Complement`](https://reference.wolfram.com/language/ref/Complement.en.md) — find the complement with respect to a universal set

[`SymmetricDifference`](https://reference.wolfram.com/language/ref/SymmetricDifference.en.md) — find the symmetric difference of any number of sets

---

[`Subsets`](https://reference.wolfram.com/language/ref/Subsets.en.md) — find all possible subsets of a set (the powerset)

[`Tuples`](https://reference.wolfram.com/language/ref/Tuples.en.md) — find all possible tuples of ``n`` elements from a set

[`Groupings`](https://reference.wolfram.com/language/ref/Groupings.en.md) — find all possible groupings of elements

---

### Testing Relations between Sets

* [`IntersectingQ`](https://reference.wolfram.com/language/ref/IntersectingQ.en.md)
* [`DisjointQ`](https://reference.wolfram.com/language/ref/DisjointQ.en.md)
* [`SubsetQ`](https://reference.wolfram.com/language/ref/SubsetQ.en.md)

### Testing Set Containment

* [`ContainsAll`](https://reference.wolfram.com/language/ref/ContainsAll.en.md)
* [`ContainsNone`](https://reference.wolfram.com/language/ref/ContainsNone.en.md)
* [`ContainsAny`](https://reference.wolfram.com/language/ref/ContainsAny.en.md)
* [`ContainsOnly`](https://reference.wolfram.com/language/ref/ContainsOnly.en.md)
* [`ContainsExactly`](https://reference.wolfram.com/language/ref/ContainsExactly.en.md)

### Testing Elements in Sets

* [`Element`](https://reference.wolfram.com/language/ref/Element.en.md)
* [`MemberQ`](https://reference.wolfram.com/language/ref/MemberQ.en.md)
* [`FreeQ`](https://reference.wolfram.com/language/ref/FreeQ.en.md)

---

### Operations on Subsets

[`SubsetCases`](https://reference.wolfram.com/language/ref/SubsetCases.en.md) — find subsets of elements matching a pattern

* [`SubsetPosition`](https://reference.wolfram.com/language/ref/SubsetPosition.en.md)
* [`SubsetCount`](https://reference.wolfram.com/language/ref/SubsetCount.en.md)

[`SubsetReplace`](https://reference.wolfram.com/language/ref/SubsetReplace.en.md) — replace subsets of elements

---

### Constructing Sets from Lists

* [`DeleteDuplicates`](https://reference.wolfram.com/language/ref/DeleteDuplicates.en.md)
* [`Sort`](https://reference.wolfram.com/language/ref/Sort.en.md)
* [`DuplicateFreeQ`](https://reference.wolfram.com/language/ref/DuplicateFreeQ.en.md)
* [`DeleteAdjacentDuplicates`](https://reference.wolfram.com/language/ref/DeleteAdjacentDuplicates.en.md)
* [`UniqueElements`](https://reference.wolfram.com/language/ref/UniqueElements.en.md)

---

### Symbolic Set Theory

* [`ForAll`](https://reference.wolfram.com/language/ref/ForAll.en.md)
* [`Exists`](https://reference.wolfram.com/language/ref/Exists.en.md)
* [`Resolve`](https://reference.wolfram.com/language/ref/Resolve.en.md)

## Related Guides

* [Logic and Boolean Algebra](https://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.en.md)
* [List Manipulation](https://reference.wolfram.com/language/guide/ListManipulation.en.md)