Sets & Relations
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Explore sets, set operations, Venn diagrams, relations, and functions - the language of modern mathematics.
Concepts in this topic
- Set Basics & Operations — Understand sets, membership, subsets, and fundamental set operations
- Venn Diagrams & Properties — Visualize set operations and learn key set identities
- Relations — Learn about binary relations and their properties
- Functions — Understand functions as special relations, injections, surjections, and bijections
Key formulas & identities
| Name | Formula | Meaning |
|---|---|---|
| Union | A ∪ B = {x | x ∈ A or x ∈ B} | The set of elements belonging to at least one of the two sets |
| Intersection | A ∩ B = {x | x ∈ A and x ∈ B} | The set of elements belonging to both sets |
| Complement | Aᶜ = {x ∈ U | x ∉ A} | The set of elements in the universal set that are not in A |
| De Morgan's Law for Sets (1) | (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ | The complement of a union is the intersection of the complements |
| De Morgan's Law for Sets (2) | (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ | The complement of an intersection is the union of the complements |
| Inclusion-Exclusion (Two Sets) | |A ∪ B| = |A| + |B| - |A ∩ B| | The size of a union equals the sum of sizes minus the intersection |
| Power Set Size | |P(A)| = 2ⁿ (n = |A|) | A set with n elements has 2ⁿ subsets |
| Cartesian Product Size | |A × B| = |A| × |B| | The number of ordered pairs is the product of the set sizes |