Q

# 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

Q. 9 Given a non-empty set X, consider the binary operation $* : P(X) \times P(X) \rightarrow P(X)$
given by $A * B = A \cap B\;\; \forall A ,B\in P(X)$, 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 ∗.

Views

Given   $* : P(X) \times P(X) \rightarrow P(X)$   is defined as  $A * B = A \cap B\;\; \forall A ,B\in P(X)$.

As we know that $A \cap X=A=X\cap A\forall A \in P(X)$

$\Rightarrow$            $A*X =A=X*A \forall A in P(X)$

Hence, X is the identity element of binary operation *.

Now, an element $A \in P(X)$ is invertible if there exists a $B \in P(X)$ ,

such that     $A*B=X=B*A$         (X is identity element)

i.e.             $A\cap B=X=B\cap A$

This is possible only if  $A=X=B$.

Hence, X is only invertible element  in $P(X)$ with respect to operation *

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