Which one of the following is not true?

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z$,define $a ^* b = a - b$.

Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.

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

On the set of positive rationals,a binary operation $*$ is defined by $a * b = \frac{2ab}{5}$. If $2 * x = 3^{-1}$,then $x = $

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 $a * b = ab^2$.