I was reading a theorem about functions:
Let $f:A\to B$ be any function. Then
$\hskip0.3in$(a) $1_B\circ f=f$.
$\hskip0.3in$(b) $f\circ 1_A=f$.
If $f$ is a one-to-one correspondence between $A$ and $B$, then
$\hskip0.3in$(c) $f^{-1}\circ f=1_A$
$\hskip0.3in$(d) $f\circ f^{-1}=1_B$
Now I am unable to decide that either the input would be from A
or B
in the part c
and d
. Previously I used to think that the function mentioned at the right always takes input from A
and the one mentioned at the left of composition takes input B
but it is not proved from part d
of the theorem. Can anyone please suggest how do we get to know what is the input?