In the set of integers $(Z, *)$,if $a * b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is:

Show that the operation $*: R \times R \rightarrow R$ defined by $a * b = a + 2b$ is not associative.

On the set of all natural numbers $N$,which one of the following $*$ is a binary operation?

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.

We define a binary relation $\sim$ on the set of all $3 \times 3$ real matrices as $A \sim B$ if and only if there exist invertible matrices $P$ and $Q$ such that $B = P A Q^{-1}$. The binary relation $\sim$ is

On the set of all non-zero reals,an operation $*$ is defined as $a * b = \frac{3ab}{2}$. In this group,a solution of $(2 * x) * 3^{-1} = 4^{-1}$ is