$.[12$ Points$]$ Let A $=\{1,2,3,4\},$ B $=\{$a$,$ b$,$ c$\},$ C $=\{$a$,$ b$,$ c$,$ d$\},$ D $=\{1,2,3\}$ and consider

the functions

f : A $->$ B f $=\{(1,$ a$),(2,$ c$),(3,$ a$),(4,$ c$)\}$

g : A $->$ C g $=\{(1,$ b$),(2,$ c$),(3,$ a$),(4,$ c$)\}$

h : C $->$ A h $=\{($a$,3),($b$,4),($c$,1),($d$,2)\}$

j : A $->$ B j $=\{(1,$ b$),(2,$ c$),(3,$ a$),(4,$ c$)\}$

m : D $->$ C m $=\{(1,$ b$),(2,$ d$),(3,$ b$)\}$

In the questions below, you are asked to select some of these functions. For each of the

functions f$,$ g$,$ h$,$ j$,$ m$,$ carefully explain why you reject$/$accept that function.

$($a$)$ Which, if any, of the functions are injective?

$($b$)$ Which, if any, of the functions are surjective?

$($c$)$ Which, if any, of the functions are bijectiv