If $*$ is a binary operation defined on $R$ by $a * b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is:

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

Show that $*: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a * b = a + 2b$ is not commutative.

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 $A^{-1} = A$.

Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.

Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?