Monics and epics

An arrow $f:a\to b$ in a category $C$ is called monic (or a monomorphism) if it is left-cancellable; i.e., if whenever $g_1,g_2:c\to a$ are arrows such that $f\circ g_1=f\circ g_2$, then $g_1=g_2$.

Dually, the arrow $f$ is called epic (or epi, or an epimorphism) if it is right-cancellable; i.e., if whenever $g_1,g_2:b\to c$ are arrows such that $g_1\circ f=g_2\circ f$, then $g_1=g_2$.

1. In $\mathbf{S}\mathbf{e}\mathbf{t}$, show that a set map $f:X\to Y$ is monic (respectively, epic) if and only if it is injective (respectively, surjective).
2. Show that, in a general category $C$, if an arrow $f:a\to b$ is an isomorphism (i.e., invertible), then $f$ is both monic and epic.
3. Show that in $\mathbf{R}\mathbf{i}\mathbf{n}\mathbf{g}$, the ring inclusion $i:\mathbf{Z}\to \mathbf{Q}$ is both monic and epic, even though the map $i$ is not surjective.