In a group $(G, *)$,for some element $a$ of $G$,if $a^{2}=e$,where $e$ is the identity element,then

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.

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 = $

In $P(X)$,the power set of a non-empty set $X$,a binary operation $*$ is defined by $A * B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?

Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^* \,3) \,^* \,4$ and $2 \,^* \,(3 \,^* \,4)$.

$^*$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$1$	$1$	$1$	$1$
$2$	$1$	$2$	$2$	$2$	$2$
$3$	$1$	$2$	$3$	$3$	$3$
$4$	$1$	$2$	$3$	$4$	$4$
$5$	$1$	$2$	$3$	$4$	$5$