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

crisp set properties - de morgan's law
(A ∪ B)’ = A’ ∩ B’ 

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

properties of crisp set- de morgan's law
(A ∩ B)’ = A’ ∪ B

Law of Contradiction:

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

properties of crisp sets on youtube

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
  • Law of Contradiction
  • Law of Excluded Middle

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