The binary operation $*$ on $R - \{-1\}$ defined by $a * b = \frac{a}{b+1}$ is:

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a - b$.

Number of binary operations on the set $\{a, b\}$ are

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.

Show that none of the operations given above has an identity element.

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $R$,define $*$ by $a * b = ab^2$.