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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.

Set Theory Calculator
∩
Define Your Sets
Enter elements separated by commas — numbers, letters, or words
⚡ Quick Examples
Elements separated by commas
Elements separated by commas
▶ Set C & Universal Set (optional)
For 3-set operations and triple Venn diagrams
Required for complement operations
⚠
✓ Set Operations Computed

Embed Set Theory Calculator Widget

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

OperationNotationDescriptionExample
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.

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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