crisp sets

What is Crisp Set

1.

A conventional set for which an element is either a member of the set or not. Learn more in: Fuzzy System Dynamics of Manpower Systems

2.

A set defined using a characteristic function that assigns a value of either 0 or 1 to each element of the universe, thereby discriminating between members and non-members of the crisp set under consideration. In the context of fuzzy sets theory, we often refer to crisp sets as “classical” or “ordinary” sets. Learn more in: Intuitionistic Fuzzy Image Processing

3.

A conventional set for which an element is either a member of the set or not. Learn more in: Fuzzy System Dynamics: An Application to Supply Chain Management

4.

A conventional set for which an element is either a member of the set or not. Learn more in: A Fuzzy Simulated Evolution Algorithm for Hard Problems

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

• (A^c)^c = A

Law of Transitivity

• If A ⊆ B, B ⊆ C, then A ⊆ C

Law of Excluded Middle

• (A ∪ A^c) = E

Law of Contradiction

• (A ∩ A^c) = Φ

De morgan laws

• (A ∪ B)^c = A^c ∩ B^c
• (A ∩ B)^c = A^c ∪ B^c