Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = a + ab$. Is the operation $^*$ commutative and associative?

Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?

On the set of positive rationals,a binary operation $*$ is defined by $a * b = \frac{2ab}{5}$. If $2 * x = 3^{-1}$,then $x = $

Consider the binary operations $^*: R \times R \rightarrow R$ and $o: R \times R \rightarrow R$ defined as $a \,^*\, b = |a-b|$ and $a \,o\, b = a$,$\forall \, a, b \in R$. Show that $^*$ is commutative but not associative,and $o$ is associative but not commutative. Further,show that $\forall \, a, b, c \in R, a \,^*\, (b \,o\, c) = (a \,^*\, b) \,o\, (a \,^*\, c)$. [If it is so,we say that the operation $^*$ distributes over the operation $o$]. Does $o$ distribute over $^*$? Justify your answer.

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.