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

- (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}

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