In a book that I read about mapping, it said:
Any mapping $f:X\rightarrow Y$ satisfies: $f1_{x}=1_{y}f=f$
$g:Y\rightarrow X$ is a reverse mapping with $f:X\rightarrow Y$ only when $gf=1_{x}$ and $fg=1_{y}$
I do not understand what $1_{x}$ and $1_{y}$ mean. Can anyone explain them to me?
$1_x:X \to X$ is the identity function that maps each element of $X$ to itself. Likewise $1_y:Y \to Y$ is the identity function that maps each element of $Y$ to itself. Thus $g$ is defined to be a reverse mapping of $f$ if composing it with $f$ on either side yields the identity. (Note that $gf$ is a function from $X$ to itself obtained by first applying $f: X \to Y$ and then $g: Y \to X$. Whereas $fg$ is a function from $Y$ to itself obtained by first applying $g:Y \to X$ and then $f:X \to Y$).