Properties of Crisp Sets

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:

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

If A is a subset of B and conversely B is a superset of A then

  • 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

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