1. Definition of a Set
- A set is a well-defined collection of distinct objects
- Objects in a set are called elements or members
- Denoted by curly braces:
{ } - Example:
A = {1, 2, 3, 4}
2. Representing Sets
- Roster Form: Listing all elements —
{red, blue, green} - Set-Builder Form: Describing properties —
{x | x is a natural number, x < 5}
3. Types of Sets
- Empty Set (∅ or { }): Set with no elements
- Finite Set: Limited number of elements
- Infinite Set: Unlimited elements (e.g., natural numbers)
- Equal Sets: Same elements
- Equivalent Sets: Same number of elements
4. Set Membership and Notation
- ∈: Element of (e.g., 2 ∈ {1, 2, 3})
- ∉: Not an element of (e.g., 5 ∉ {1, 2, 3})
- Cardinality (|A|): Number of elements in a set
5. Subsets
- Subset (⊆): All elements of A are in B — A ⊆ B
- Proper Subset (⊂): A ⊆ B but A ≠ B
- Superset (⊇): B contains all elements of A
- Every set is a subset of itself
- Empty set is a subset of every set
6. Universal Set and Power Set
- Universal Set (U): Set containing all elements under consideration
- Power Set (P(A)): Set of all subsets of A
- If |A| = n, then |P(A)| = 2ⁿ
7. Set Operations
- Union (A ∪ B): All elements in A or B or both
- Intersection (A ∩ B): Elements in both A and B
- Difference (A – B or A \ B): Elements in A but not in B
- Complement (A’): Elements not in A (with respect to U)
- Cartesian Product (A × B): All ordered pairs (a, b) where a ∈ A, b ∈ B
8. Venn Diagrams
- Visual representation of sets and their relationships
- Circles represent sets
- Rectangle represents universal set
- Used to show unions, intersections, and complements
9. Disjoint Sets
- Sets with no common elements
- A ∩ B = ∅
10. Basic Set Identities
- Commutative: A ∪ B = B ∪ A | A ∩ B = B ∩ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- De Morgan’s: (A ∪ B)’ = A’ ∩ B’ | (A ∩ B)’ = A’ ∪ B’
Rules/laws of Set Theory:
1. Laws of Equality
- Reflexive: A = A
- Symmetric: If A = B, then B = A
- Transitive: If A = B and B = C, then A = C
2. Laws of Subset
- Reflexive: A ⊆ A
- Antisymmetric: If A ⊆ B and B ⊆ A, then A = B
- Transitive: If A ⊆ B and B ⊆ C, then A ⊆ C
- Empty set is subset of every set: ∅ ⊆ A
3. Identity Laws
- A ∪ ∅ = A
- A ∩ U = A
- A ∩ ∅ = ∅
- A ∪ U = U
4. Complement Laws
- A ∪ A’ = U
- A ∩ A’ = ∅
- (A’)’ = A
- U’ = ∅
- ∅’ = U
5. Idempotent Laws
- A ∪ A = A
- A ∩ A = A
6. Commutative Laws
- A ∪ B = B ∪ A
- A ∩ B = B ∩ A
7. Associative Laws
- (A ∪ B) ∪ C = A ∪ (B ∪ C)
- (A ∩ B) ∩ C = A ∩ (B ∩ C)
8. Distributive Laws
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
9. De Morgan’s Laws
- (A ∪ B)’ = A’ ∩ B’
- (A ∩ B)’ = A’ ∪ B’
10. Absorption Laws
- A ∪ (A ∩ B) = A
- A ∩ (A ∪ B) = A
11. Difference Laws
- A – B = A ∩ B’
- A – A = ∅
- A – ∅ = A
- ∅ – A = ∅