Properties of crisp set helps us to simplify many mathematical set operations. Crisp set are collection of unordered, district elements. We can perform various crisp set operations on the crisp set. It is recommended to reader to first navigate through the crisp set operations for better understanding of properties of crisp set.

Crisp set possesses following properties. We will demonstrate each of them with suitable example.

We will be using following sets for the further discussion:

X = {1, 2, 3, 4, 5, 6}

A = {1, 2, 3},

B = {2, 3, 4},

C = {5, 6}

### Involution:

Involution states that the complement of complement of set A would be set A it self.

For given data,

A’ = X – A = {4, 5, 6}

(A’)’ = X – A’ = {1, 2, 3} = A

### Commutativity:

Commutativity property states that the operation can be performed irrespective of order of the operand. For example, addition is commutative operator, so 2 + 3 or 3 + 2 yields the same result. But, subtraction is not commutative, so 3 – 2 ≠ 2 – 3.

Proving union is commutative:

A ∪ B = {1, 2, 3, 4}  → LHS

B ∪ A = {1, 2, 3, 4}  → RHS

Proving intersection is commutative:

A ∩ B = {2, 3}  → LHS

B ∩ A = {2, 3} → RHS

### Associativity:

Associativity property allows us to perform the operations by grouping the operands and keeping them in similar order.

(A ∪ B) ∪ C= A ∪ ( B ∪ C )

For given data:

A ∪ B = {1, 2, 3, 4}

(A ∪ B) ∪ C={1, 2, 3, 4, 5, 6} → LHS

B ∪ C = {2, 3, 4, 5, 6}

A ∪ (B ∪ C) = {1, 2, 3, 4, 5, 6} → RHS

(A ∩ B) ∩ C= A ∩ ( B ∩ C)

For given data:

A ∩ B = {2, 3}

(A ∩ B) ∩ C = ϕ → LHS

B ∩ C = ϕ

A ∩ (B ∩ C) = ϕ → RHS

### Distributivity:

Mathematically it is defined as,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

B ∩ C = ϕ

A ∪ (B ∩ C) = {1, 2, 3}→LHS

A ∪ B = {1, 2, 3, 4}

A ∪ C = {1, 2, 3, 5, 6}

(A ∪ B) ∩ (A ∪ C) = {1, 2, 3} → RHS

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

B ∪ C = {2, 3, 4, 5, 6}

A ∩ (B ∪ C) = {2, 3} → LHS

A ∩ B = {2, 3}

A ∩ C = ϕ

(A ∩ B) ∪ (A ∩ C) = {2, 3} → RHS

### Absorption:

Mathematically absorption is defined as,

A ∪ (A ∩ B) = A

For the given data:

A ∩ B = {2, 3}

A ∪ (A ∩ B) = {1, 2, 3} = A

A ∩ (A ∪ B) = A

For the given data:

A ∪ B = {1, 2, 3, 4}

A ∩ (A ∪ B) = {1, 2, 3} = A

### Idempotency/Tautology:

Idempotency is defined as,

A ∪ A = A

A ∩ A = A

For the given data,

A ∪ A = {1, 2, 3} = A

A ∩ A = {1, 2, 3} = A

### Identity:

Mathematically, we can define this property as,

A ∪ X = X

A ∩ X = A

A ∪ ϕ = A

A ∩ ϕ = ϕ

For the given data,

A ∪ X = {1, 2, 3, 4, 5, 6} = X

A ∩ X = {1, 2, 3} = A

A ∪ ϕ = {1, 2, 3} = A

A ∩ ϕ = { } = ϕ

### De Morgan’s Laws:

Mathematically, De Morgan’s laws are defined as,

Law 1: (A ∪ B)’ = A’ ∩ B’

For the given data:

A ∪ B = {1, 2, 3, 4}

(A ∪ B)’ = {5, 6}  → LHS

A’ = {4, 5, 6}

B’ = {1, 5, 6}

A’ ∩ B’ = {5, 6} = (A ∪ B)’ → RHS

Law 2: (A ∩ B)’ = A’ ∪ B’

For the given data:

A ∩ B = {2, 3}

(A ∩ B)’ = {1, 4, 5, 6} → LHS

A’ = {4, 5, 6}

B’ = {1, 5, 6}

A’ ∪ B’ = {1, 4, 5, 6} = (A ∩ B)’ → RHS

Mathematically it is defined as,

A ∩ A’ = ϕ

For the given data:

A’ = {4, 5, 6}

A ∩ A’ = { } = ϕ

### Law of Excluded Middle:

Mathematically it is defined as,

A ∪ A’ = X

For the given data:

A’ = {4, 5, 6}

A ∪ A’ = {1, 2, 3, 4, 5, 6} = X

## Test Your Knowledge:

Let X = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 5}, B = {1, 3, 4, 6, 7}, C = {2, 4, 6, 8}. Prove following properties for given sets.

• Involution
• Commutativity
• Associativity
• Distributivity
• Absorption
• Idempotency
• Identity
• De Morgan’s Laws