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:

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.

Show that none of the operations given above has an identity element.

Let $P$ be the set of all subsets of a given set $X$. Show that $\cup: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cup B$ and $\cap: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cap B$ are binary operations on the set $P$.

Which of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?