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

Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.

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

In the group $G = \{1, 2, 3, 4, 5, 6\}$ under $\otimes_{7}$,the solution of $4 \otimes_{7} x = 5$ is

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.

State whether the following statement is true or false and justify your answer: If $^*$ is a commutative binary operation on $N$,then $a ^* (b ^* c) = (c ^* b) ^* a$.