Set Theory Guide
Set Notation Reference
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| ∪ | Union | Elements in A or B | {1,2}∪{2,3}={1,2,3} |
| ∩ | Intersection | Elements in both A and B | {1,2}∩{2,3}={2} |
| ∁ | Complement | Elements not in A | ∁ᵤA |
| ∖ | Difference | Elements in A but not B | {1,2,3}∖{2}={1,3} |
| ⊆ | Subset | A is subset of B | {1,2}⊆{1,2,3} |
| ⊂ | Proper Subset | A⊆B and A≠B | {1,2}⊂{1,2,3} |
| ∅ | Empty Set | Set with no elements | {}=∅ |
| ∈ | Element of | x is in set A | 2∈{1,2,3} |
| ∉ | Not element of | x is not in set A | 5∉{1,2,3} |
| |A| | Cardinality | Number of elements | |{a,b,c}|=3 |
| × | Cartesian Product | All ordered pairs (a,b) | A×B |
| 𝒫(A) | Power Set | All subsets of A | 𝒫({1,2})={{"{∅,{1},{2},{1,2}}"}} |
De Morgan's Laws
Law 1
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
Complement of union = intersection of complements
Law 2
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
Complement of intersection = union of complements
Inclusion-Exclusion Principle
|A ∪ B| = |A| + |B| - |A ∩ B|
|A ∪ B ∪ C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|