$(15$ 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.