Set Theory and Sets are one of the fundamental and widely present concepts in mathematics. A Crisp Setor simple a Set is a well-defined collection of distinct objects where each object is considered in its own right. Here are the 11 main properties/laws of crisp sets:
If A is a subset of B and conversely B is a superset of A then
Law of Commutativity:
- (A ∪ B) = (B ∪ A)
- (A ∩ B) = (B ∩ A)
Law of Associativity:
- (A ∪ B) ∪ C = A ∪ (B ∪ C)
- (A ∩ B) ∩ C = A ∩ (B ∩ C)
Law of Distributivity:
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Idempotent Law
- A ∪ A = A
- A ∩ A = A
Identity Law
- A ∪ Φ = A => A ∪ E = E
- A ∩ Φ = Φ => A ∩ E = A
Here Φ is empty set and E is universal set or universe of discourse.
Law of Absorption
- A ∪ (A ∩ B) = A
- A ∩ (A ∪ B) = A
Involution Law
- (Ac)c = A
Law of Transitivity
- If A ⊆ B, B ⊆ C, then A ⊆ C
Law of Excluded Middle
- (A ∪ Ac) = E
Law of Contradiction
- (A ∩ Ac) = Φ
De morgan laws
- (A ∪ B)c = Ac ∩ Bc
- (A ∩ B)c = Ac ∪ Bc