Question

Let $f:A\to B,g:B\to A$ be two functions such that $fog=I_B$, then which among the following is/are true

• A. $f$ is a surjective function
• B. $g$ is an injective function.
• C. $g$ is many-one function
• D. $f$ is an into function.

Answer 1

Answer: (A), (B)

$g$ is an injective function.

Let $b$ be an arbitrary element of $B$, $\begin{array}{c}\text{Let}g\left(b\right)=a,a\in A\\ \Rightarrow f\left(a\right)=f\left(g\right(b\left)\right)\\ =fog\left(b\right)\\ =I_A\left(b\right)\end{array}$
$f\left(a\right)=b$
$\therefore$ For every $b\in B,$ there exists $a\in A,$ such that $f\left(a\right)=b$
So, $f$ is surjective function
Let $x,y$ be any two elements of $B$ such that
$g\left(x\right)=g\left(y\right)\Rightarrow f\left(g\right(x\left)\right)=f\left(g\right(y\left)\right)$ $\Rightarrow fog\left(x\right)=\mathrm{fog}\left(y\right)$
$I_B\left(x\right)=I_B\left(y\right)$
$x=y$
$\therefore g$ is injective function