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(B) The binary operation is defined as $A * B = A \cup B$ for all $A, B \in P(X)$.$1$. Commutative Law: $A * B = A \cup B = B \cup A = B * A$. Thus,it

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The inverse law is not satisfied.

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(B) The binary operation is defined as $A * B = A \cup B$ for all $A, B \in P(X)$.$1$. Commutative Law: $A * B = A \cup B = B \cup A = B * A$. Thus,it is commutative.$2$. Associative Law: $(A * B) * C = (A \cup B) \cup C = A \cup (B \cup C) = A * (B * C)$. Thus,it is associative.$3$. Identity Law: The identity element $E$ must satisfy $A * E = A$,which means $A \cup E = A$. This holds for $E = \phi$. Thus,the identity element exists.$4$. Inverse Law: For an element $A$ to have an inverse $B$,we must have $A * B = E$,where $E = \phi$. This implies $A \cup B = \phi$. This is only possible if $A = \phi$ and $B = \phi$. For any non-empty set $A 
eq \phi$,there is no $B \in P(X)$ such that $A \cup B = \phi$. Therefore,the inverse law is not satisfied.

In $P(X)$,the power set of a non-empty set $X$,a binary operation $$ is defined by $A B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?

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In $P(X)$,the power set of a non-empty set $X$,a binary operation $*$ is defined by $A * B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?