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🔲 Set Theory - Complete Foundation Guide

Comprehensive exploration of Sets, Union, Intersection, Complement, and Set Operations

📚 Set Theory Basics

Set: A collection of distinct objects

Element: An object in a set (a ∈ A)

Empty Set: ∅ = {} (contains no elements)

Universal Set: U (contains all elements under consideration)

Subset: A ⊆ B (all elements of A are in B)

Set Operations:

  • Union: A ∪ B (elements in A or B or both)
  • Intersection: A ∩ B (elements in both A and B)
  • Complement: A' (elements not in A)
  • Difference: A - B (elements in A but not B)
  • Symmetric Difference: A ⊕ B = (A - B) ∪ (B - A)

🧮 Set Operations Calculator

Perform operations on two sets:

🎯 Subset & Membership Checker

Check relationships between sets and elements:

📊 Venn Diagram Info

Analyze set relationships and cardinalities:

📝 Worked Examples

Example 1: Union

Given: A = {1,2,3}, B = {3,4,5}

Find: A ∪ B

Solution: Combine all elements, no duplicates

Answer: A ∪ B = {1,2,3,4,5}

Example 2: Intersection

Given: A = {1,2,3,4}, B = {3,4,5,6}

Find: A ∩ B

Solution: Elements common to both sets

Answer: A ∩ B = {3,4}

Example 3: Complement

Given: U = {1,2,3,4,5}, A = {1,3,5}

Find: A'

Solution: Elements in U but not in A

Answer: A' = {2,4}

Example 4: De Morgan's Law

Law: (A ∪ B)' = A' ∩ B'

Given: A = {1,2}, B = {2,3}, U = {1,2,3,4}

Left side: (A ∪ B)' = {1,2,3}' = {4}

Right side: A' ∩ B' = {3,4} ∩ {1,4} = {4}

Verified: Both sides equal {4} ✓