Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = (a - b)^{2}$ for all $a, b \in Q$ is commutative.

Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^* \,3) \,^* \,4$ and $2 \,^* \,(3 \,^* \,4)$.

$^*$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$1$	$1$	$1$	$1$
$2$	$1$	$2$	$2$	$2$	$2$
$3$	$1$	$2$	$3$	$3$	$3$
$4$	$1$	$2$	$3$	$4$	$4$
$5$	$1$	$2$	$3$	$4$	$5$

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a ^* b = a - b$. Determine whether the operation $^*$ is commutative and associative.

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?

If $a * b = 10ab$ on $Q^{+}$,then find the inverse of $0.01$.

Let $P$ be the set of all subsets of a given set $X$. Show that $\cup: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cup B$ and $\cap: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cap B$ are binary operations on the set $P$.