HW $2$ Problem $2.$ Given a map $$:$xY,$ show the following statements:

$$ is injective if and only if there is some map $$:$(x)x$ such that $$@$$ is the identity

map of $x$;

$$ is surjective if and only if there is some map $$:$Yx$ such that $$@$$ is the identity

map of $Y$;

$$ in Statement $1$ is unique per $$ if it exists.

Remark. In the above problem, we used the following notation:

$(x)$:$=\{(x)$:xinx$\},$

which is a subset of $Y.$

We say that a map is bijective if it is injective and surjective.