For any two real numbers,an operation $*$ defined by $a * b = 1 + ab$ is

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

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

Given a non-empty set $X$,consider the binary operation $^*: P(X) \times P(X) \rightarrow P(X)$ defined by $A \,^*\, B = A \cap B$ for all $A, B \in P(X)$,where $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P(X)$ with respect to the operation.

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a \,^* \,b = a^{2} + b^{2}$. Which of the following is true?

Which of the following is not a group with respect to the given operation?

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