On the set $Z$ of all integers,the operation $*$ is defined by $a * b = a + b - 5$. If $2 * (x * 3) = 5$,then $x$ is equal to:

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 $a * b = ab^2$.

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = a + ab$. Is the operation $^*$ commutative and associative?

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 $a * b = |a - b|$.

If the operation $ \oplus $ is defined by $ a \oplus b = a^{2} + b^{2} $ for all real numbers $ a $ and $ b $,then $ (2 \oplus 3) \oplus 4 = $

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 $R$.