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Given a non-empty set X, let * : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set ϕ is the identity for the operation * and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – ϕ) ∪ (ϕ– A) = A and (A – A) ∪ (A – A) = A * A = ϕ).
Given a non-empty set X, let * : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set ϕ is the identity for the operation * and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – ϕ) ∪ (ϕ– A) = A and (A – A) ∪ (A – A) = A * A = ϕ).
CBSE | Class 12 | Maths | Miscellaneous | NCERT | Relations and Functions
Given a non-empty set X, let * : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set ϕ is the identity for the operation * and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – ϕ) ∪ (ϕ– A) = A and (A – A) ∪ (A – A) = A * A = ϕ).

solution:

x ∈ P(x)

Furthermore,

ϕ is the character component for the activity * on P(x). Additionally A*A=

=

Each component An of P(X) is invertible with A-1 = A.

i

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