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

Let $*^{\prime}$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$. Is the operation $*^{\prime}$ same as the operation $*$ defined in Exercise $4$ above? Justify your answer.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = ab + 1$.

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 $P$.

Let $^*$ be the binary operation on $N$ given by $a \, ^* \, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ commutative?