Let $f$ be a total function on some nonempty set $D$. In the following propositions, x and y are variables ranging over $D$, and g is a variable ranging over total functions on $D$. Indicate all of the propositions that are equivalent to the proposition that $f$ is an injection.
(1) $\exists g\,\forall x\ : (g \circ f) (x) = x $
(2) $\exists g\,\forall x\ : (f \circ g) (x) = x $
I know that (1) is true and (2) is false, but I don't understand why. Could someone please explain? Thank you.
Hint 1
Assume that $f(x)=f(y)$ with $x\ne y.$ Then
$$x=g(f(x))=g(f(y))=y$$ gives a contradiction.