For each of the following statements, determine whether it is true or false and then prove it $($if it is true$)$ or disprove it $($if it is false$).$

$[$What we expect: For each part, first state if the statement is true or false. Then present a proof. You do not need to present your proof in formal logic format. An informal proof which is clear and correct would be sufficient.$]$

$3.$a$.[1$ pts$]$ n$^2$ n $+41$ is prime for every non$-$negative integer n$.$

$3.$b$.[2$ pts$]$ The ratio $($result of division$)$ of any two non$-$zero rational numbers is rational.

$[$HINT: Formally speaking, you must prove that

$$x$$y$($x $=0$y $=0$p$1$q$1($q$1=0$x $=$ p$1/$q$1)$p$2$q$2($q$2=0$y $=$ p$2/$q$2)->$p$3$q$3($q$3=0$x$/$y $=$ p$3/$q$3)]$

$3.$c$.[1$ pts$]$ Let f be a real$-$valued function of the real numbers $($i$.$e$.,$ f : R $->$ R$).$ f$($x$)$ is irrational if and only if x is irrational.

$3.$d$.[2$ pts$]$ If a and b are irrational then a$^$b is also irrational.