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.

(A) binary operation $^*$ on $\{1, 2\}$ is a function from $\{1, 2\} \times \{1, 2\}$ to $\{1, 2\}$,i.e.,a function from $\{(1, 1), (1, 2), (2, 1), (2, 2)\}$ to $\{1, 2\}$.
Since $1$ is the identity for the binary operation $^*$,we must have $1 * 1 = 1$,$1 * 2 = 2$,and $2 * 1 = 2$.
This determines the values for the pairs $(1, 1)$,$(1, 2)$,and $(2, 1)$.
For the pair $(2, 2)$,we are given that $2$ is the inverse of $2$. By definition of inverse,$2 * 2$ must equal the identity element,which is $1$.
Thus,$2 * 2 = 1$.
Since all values of the function $^*$ are uniquely determined,there is exactly one such binary operation.