If $A = \{a, b, c\}$,then the number of binary operations on $A$ is

Let $*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = \frac{ab}{4}$. Which of the following is true?

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.

Show that the number of binary operations on $\{1, 2\}$ having $1$ as identity and having $2$ as the inverse of $2$ is exactly one.

Which of the following statements is false?

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.