Questions

$(10$ points$)$ Let $x=\{1,3,5,7,9\}$ and $Y=\{2,4,6,8,10\}.$

$($a$)$ Define $f$:$xY$ by specifying that

$f(1)=8,$

$f(3)=6$

$f(5)=4$

$f(7)=8$

$f(9)=10.$

Is $f$ one$-$to$-$one? Is $f$ onto? Explain your answers.

$($b$)$ Define $g$:$xY$ by specifying that:

$g(1)=2$

$g(3)=4$

$g(5)=6$

$g(7)=8$

$g(9)=10$

Is $g$ one$-$to$-$one? Is $g$ onto? Explain your answers.

$(15$ 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 $F$

$($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?