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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.

Q. 13 Given a non-empty set X, let $* : P(X) \times P(X) \rightarrow P(X)$ be defined as
$A * B = (A - B) \cup (B -A), \;\forall A, B \in P(X).$ Show that the empty set $\phi$ is the
identity for the operation ∗ and all the elements A of P(X) are invertible with
$A^{-1} =A$ . (Hint : $(A - \phi) \cup (\phi - A) = A$ and $(A -A) \cup (A - A) = A * A = \phi$ ).

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S seema garhwal

Let $* : P(X) \times P(X) \rightarrow P(X)$ be defined as $A * B = (A - B) \cup (B -A), \;\forall A, B \in P(X).$

Let $A \in P(X)$ . Then

$A * \phi = (A - \phi ) \cup (\phi -A) = A\cup \phi =A$

$\phi * A = (\phi - A ) \cup (A -\phi ) = \phi \cup A =A$

$\therefore \, \, \, A*\phi =A= \phi * A$ for all $A \in P(X)$

Thus, $\phi$ is identity element for operation *.

An element $A \in P(X)$ will be invertible if there exists $B\in P(X)$ ,

such that $A*B=A=B*A$ . (here $\phi$ is identity element)

$A * A = (A - A ) \cup (A -A) = \phi \cup \phi =\phi$ $\forall A \in P(X)$

Hence, all elements A of P(X) are invertible with $A^{-1}=A.$

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