$(20$ points$)$ Show that the following arguments are valid.

$($a$)$ Premises: $A(B^{{?}^{?}}C),A$

Conclusion: $B$

$($b$)$ Premises: $UC,$LvvU,notL

Conclusion: $C$

$($c$)$ Premises: $(A^{{?}^{?}}B)(C^{{?}^{?}}D),A^{{?}^{?}}C,B$

Conclusion: $D$

$($d$)$ Premises: $(notPvvQ)R,R(SvvT),not{S}^{{?}^{?}}$notU,notU$$notT

Conclusion: $P$

$(10$ points$)$ Show that the following arguments are valid.

$($a$)$ Premises: EEx$(A(x{)}^{{?}^{?}}B(x)),$AAx$(A(x)C(x))$

Conclusion: EEx$(C(x{)}^{{?}^{?}}B(x))$

$($b$)$ Premises: EEx$(notK(x{)}^{{?}^{?}}$notH$(x))vvAAy(L(y{)}^{{?}^{?}}$notL$(y))$

Conclusion: EExnotH$(x)$

$(10$ points$)$ Prove or disprove that the product of a nonzero rational number and an

irrational number is irrational

$(10$ points$)$ Prove using contrapositive: If $4k^2+3k+2$ is odd then $k$ is odd.

$(10$ points$)$ Prove that for all ninN,

$1^2+4^2+7^2+{10}^2+$cdots$+(3n-2{)}^2=\frac{n(6n^2-3n-1)}{2}$

$(10$ points$)$ Show $n^3+2n$ is divisible by $3,$AAninN.

$(0$ points$)$ Show that it is possible to arrange the numbers $1,2,$dots,$n$ in a row so that

the average of any two of these numbers never appears between them.