Given a non-empty set X, consider the binary operation * : P(X) * P(X)

1.	Given a non-empty set X, consider the binary operation * : P(X) * P(X) → P(X) given by A * B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.
Answer» Let E ∈ P (X) be an identity elements, then A * E = E * A = A ∀ A ∈ P(X) ⇒ A ∩ E = E ∩ A = A ∀ A ∈ P(X) ⇒ X ∩ E = X as X ∈ P (X) ⇒ X C E Also E C X as E ∈ P (X) ∴ E = X Thus, X is the identity element. Let A ∈ P(X) be invertible, then there exists B ∈ P(X) such that A * B = B * A = X, the identity element. ⇒ A ∩ B = B ∩ A = X X C A & X C B Also A, B, ⊂ X as A, B ∈ P (X) A = X = B X is the only invertible element and – X₁ = B = X.

Given a non-empty set X, consider the binary operation * : P(X) * P(X) → P(X) given by A * B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.

Answer»

Let E ∈ P (X) be an identity elements, then

A * E = E * A = A ∀ A ∈ P(X)

⇒ A ∩ E = E ∩ A = A ∀ A ∈ P(X)

⇒ X ∩ E = X as X ∈ P (X)

⇒ X C E Also E C X as E ∈ P (X)

∴ E = X

Thus, X is the identity element.

Let A ∈ P(X) be invertible, then there exists

B ∈ P(X) such that A * B = B * A = X, the identity element.

⇒ A ∩ B = B ∩ A = X X C A & X C B

Also A, B, ⊂ X as A, B ∈ P (X)

A = X = B

X is the only invertible element and – X₁ = B = X.