# If $A=\{1,2\},$ then how many binary operations having 1 as identity and 2 as inverse of 2 are possible?

(A) 1 (B) 2 (C) 4 (D) 8

Toolbox:
• A Binary Operation on a non empty set $A$ is a function from $A\times A\rightarrow A$
$Given:\:\:A=\{1,2\}$
Binary operation $\large *$ is a function from $\{(1,1),(1,2),(2,1),(2,2)\}\rightarrow \{1,2\}$
Also given that 1 is identity element and 2 is inverse of 2.
$\Rightarrow\:\large 1 *1=1,\:\:1\large *2=2\large *1=2\:\:and\:\:2\large *2=1$
This is the only possible mapping for the given conditions.
$\therefore \:$ only one binary operation is possible.