$(15$ points$)$ Let $x=\{a,b,c,d\}$ and $Y=\{e,f,g\}.$ Define functions $F$ and $G$ by the arrow diagrams below.

Domain of $F$

Co$-$domain of $F$

Domain of $G,$ Co$-$domain of $G$

$($a$)$ Is F one$-$to$-$one? Why or why not? Is it onto? Why or why not?

$($b$)$ Is G one$-$to$-$one? Why or why not? Is it onto? Why or why not?

$2$

$($c$)$ Redefine F and G to make them one$-$to$-$one correspondences. Explain your answer.