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

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.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

Show that subtraction and division are not binary operations on the set of natural numbers $N$.

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

Let $A = N \times N$ and $^*$ be the binary operation on $A$ defined by $(a, b) \,^*\, (c, d) = (a + c, b + d)$. Determine whether the operation $^*$ is commutative,associative,and has an identity element.