# Properties of crisp set: All at one place

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

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

We will be using the following sets for 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 itself.

For the given data,

A’ = X – A = {4, 5, 6}

(A’)’ = X – A’ = {1, 2, 3} = A

### Commutativity:

The commutativity property states that the operation can be performed irrespective of the order of the operand. For example, addition is a 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:

The 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

## Watch on YouTube: Properties of crisp set

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 the following properties for given sets.

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