The set $\{-1, 0, 1\}$ is not a multiplicative group because of the failure of

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?

State whether the following statement is true or false. Justify: For an arbitrary binary operation $^*$ on a set $N$,$a \,^* \,a = a$ for all $a \in N$.

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = (a - b)^2$. Determine whether the operation is commutative and associative.

Let $*^{\prime}$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$. Is the operation $*^{\prime}$ same as the operation $*$ defined in Exercise $4$ above? Justify your answer.

Which of the following is false?