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Question
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in 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*.
Solution
It is given that .:`P(X) x P(X) -> P(X)` is defined as `A * B = A "∩" B ∀ A, B in P(X) `
We know that A ∩ X = A = X ∩ A ∀ A ∈ P(X)
`=> A * X = A = X * A ∀ A ∈ P(X)`
Thus, X is the identity element for the given binary operation *.
Now, an elementis A ∈ P(X)invertible if there exists B ∈ P(X) such that
A * B = X = B * A (As X is the identity element)
i.e
A ∩ B = X = B ∩ A
This case is possible only when A = X = B.
Thus, X is the only invertible element in P(X) with respect to the given operation*.
Hence, the given result is proved.
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