What is a set in mathematics?

A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by capital letters (A, B, C) and their elements are listed inside curly braces, e.g., A = {1, 2, 3}. Sets can contain numbers, letters, or any other objects.

What is the difference between union and intersection?

Union (A ∪ B) combines all elements from both sets — an element is in the union if it is in A OR in B. Intersection (A ∩ B) contains only elements common to both sets — an element is in the intersection only if it is in A AND in B.

What is a Cartesian product?

The Cartesian product A × B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. For example, if A = {1, 2} and B = {x, y}, then A × B = {(1,x), (1,y), (2,x), (2,y)}. The size of the Cartesian product is |A| × |B|.

The power set ℘(A) is the set of all possible subsets of A, including the empty set and A itself. If A has n elements, the power set has 2^n elements. For example, ℘({1,2}) = {∅, {1}, {2}, {1,2}}.

Set Theory Calculator

Perform set operations including Union (A ∪ B), Intersection (A ∩ B), Difference (A − B), Symmetric Difference (A ∆ B), Cartesian Product (A × B), Power Set, and Complement. Visualize with interactive Venn diagrams.

Free to useNo sign-up requiredUpdated Feb 2026

Set Theory CalculatorTry it now — free ▼

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Define Your Sets

Enter elements separated by commas — numbers, letters, or words

⚡ Quick Examples

A Set A

Elements separated by commas

B Set B

Elements separated by commas

▶ Set C & Universal Set (optional)

C Set C (optional)

For 3-set operations and triple Venn diagrams

U Universal Set

Required for complement operations

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✓ Set Operations Computed

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About Set Theory Calculator

What Is Set Theory?

Set theory is a branch of mathematical logic that studies collections of objects called sets. Founded by Georg Cantor in the 1870s, it has become the foundation of virtually all modern mathematics. A set is defined by its members — two sets are equal if and only if they have exactly the same elements.

Discrete Mathematics — the basis for combinatorics, graph theory, and formal languages
Computer Science — data structures (HashSet, TreeSet), database queries (JOIN = intersection, UNION = union), and type systems
Probability — events are modeled as sets, with union and intersection corresponding to OR and AND events
Logic — Venn diagrams visualize logical relationships; set operations mirror logical operators

How to Use This Set Theory Calculator

Enter the elements of each set separated by commas. You can use numbers, letters, words, or any text as elements. The calculator will automatically compute all major set operations and display interactive Venn diagrams.

Type elements separated by commas — e.g., 1, 2, 3, 4, 5 or apple, banana, cherry
Use Set C (optional) for three-set operations and triple Venn diagrams
Define a Universal Set to calculate complements (Aᶜ, Bᶜ)
Click on the Venn diagram operation buttons to highlight different regions
Use the Properties tab to check cardinality, subset relationships, and set equality

Set Operations Reference

Operation	Notation	Description	Example
Union	A ∪ B	Elements in A or B (or both)	`{1,2,3} ∪ {3,4,5} = {1,2,3,4,5}`
Intersection	A ∩ B	Elements in both A and B	`{1,2,3} ∩ {3,4,5} = {3}`
Difference	A − B	Elements in A but not in B	`{1,2,3} − {3,4,5} = {1,2}`
Symmetric Difference	A ∆ B	Elements in A or B but not both	`{1,2,3} ∆ {3,4,5} = {1,2,4,5}`
Cartesian Product	A × B	All ordered pairs (a,b) where a∈A, b∈B	`{1,2} × {a,b} = {(1,a),(1,b),(2,a),(2,b)}`
Power Set	℘(A)	All possible subsets of A	`℘({1,2}) = {∅,{1},{2},{1,2}}`
Complement	Aᶜ	Elements in Universal set but not in A	`If U={1,2,3,4,5}, A={1,2} → Aᶜ={3,4,5}`
Is Subset	A ⊆ B	Whether every element of A is also in B	`{1,2} ⊆ {1,2,3} = True`

Key Set Theory Laws

These fundamental laws govern how set operations interact, similar to the laws of algebra for numbers:

Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A
Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan's Laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
Identity: A ∪ ∅ = A and A ∩ U = A
Complement: A ∪ Aᶜ = U and A ∩ Aᶜ = ∅
Idempotent: A ∪ A = A and A ∩ A = A

Applications of Set Theory

Understanding set operations is crucial in many fields:

SQL Databases — UNION, INTERSECT, EXCEPT are set operations on query results
Python Programming — the set type supports | (union), & (intersection), - (difference)
Probability Theory — P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (inclusion-exclusion principle)
Digital Logic — set operations correspond to logic gate operations (OR, AND, NOT)
Data Analysis — comparing datasets, finding common records, identifying unique entries

Reference this content, page, or tool as:

"Set Theory Calculator" at https://MiniWebtool.com/set-theory-calculator/ from MiniWebtool, https://MiniWebtool.com/

Set Theory Calculator uses standard set theory definitions. For more information, see Set theory - Wikipedia.

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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