In the group $(G, \times_{15})$,where $G = \{3, 6, 9, 12\}$ and $\times_{15}$ is multiplication modulo $15$,the identity element is

Show that $*: R \times R \rightarrow R$ given by $(a, b) \rightarrow a+4 b^{2}$ is a binary operation.

Number of binary operations on the set $\{a, b\}$ are

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

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.