Set Theory Calculators

Comprehensive collection of set operations for mathematical set theory

Basic Set Operations

Union (A ∪ B)

All elements from both sets without duplicates - the "total" of both sets

Intersection (A ∩ B)

Common elements of both sets - what both sets share

Difference (A \ B)

Elements from A that are not in B - "A without B"

Advanced Set Operations

Symmetric Difference (A ⊕ B)

Exclusive or - elements that are in only one set (XOR operation)

Complement (A^c)

All elements of the universal set that are not in A

About Set Theory

Set theory is a fundamental branch of mathematics that deals with the study of collections of objects. Set operations form the foundation for:

Databases - SQL operations
Logic - Boolean algebra
Probability - Event spaces

Computer Science - Algorithms
Statistics - Data analysis
Combinatorics - Counting problems

Important Properties of Set Operations

Commutative Laws

A ∪ B = B ∪ A
A ∩ B = B ∩ A

Associative Laws

(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Laws

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De Morgan's Laws

(A ∪ B)^c = A^c ∩ B^c
(A ∩ B)^c = A^c ∪ B^c

Tip: Use Venn diagrams to visualize set operations. Union shows "or" (∪), intersection shows "and" (∩), and symmetric difference shows "exclusive or" (⊕).

Practical Application Examples

Database Operations

UNION: Combining query results
INTERSECT: Finding common records
EXCEPT: Difference between tables

Programming

Set.union(): Merging collections
Set.intersection(): Finding overlaps
Set.difference(): Identifying differences

Probability Theory

P(A ∪ B): Probability of A or B
P(A ∩ B): Probability of A and B
P(A \ B): Conditional probabilities

Logic and Boolean Algebra

OR Gate: Corresponds to union
AND Gate: Corresponds to intersection
XOR Gate: Corresponds to symmetric difference

Quick Reference

A ∪ B

Union

A ∩ B

Intersection

A \ B

Difference

A^c

Complement

A ⊕ B

Symmetric Difference