Question $9(15$ points$)$

Prove that the following wff are valid arguments

$($a$)(EEx)[P(x{)}^{{?}^{?}}Q(x){]}^{{?}^{?}}(AAy)[Q(y)R(y)](EEx)[P(x{)}^{{?}^{?}}R(x)]$

$($b$)(AAx)(AAy)[(P(x{)}^{{?}^{?}}S(x,y))Q(y){]}^{{?}^{?}}(EEx)B(x{)}^{{?}^{?}}(AAx)(B(x)P(x){)}^{{?}^{?}}(AAx)(EEy)S(x,y)$

$(EEx)Q(x)$

$($c$)(AAx)[S(x)(EEy)[P(x,y{)}^{{?}^{?}}T(y)]{]}^{{?}^{?}}(EEx)[C(x{)}^{{?}^{?}}S(x)](EEx)(EEy)[C(x{)}^{{?}^{?}}T(y{)}^{{?}^{?}}P(x,y)]$

Question $10(20$ points$)$

Prove that the following arguments are true by using predicate logic. Use the predicate symbols given.

$($a$)$ Every computer science student works harder than somebody, and everyone who works

harder than any other person gets less sleep than that person. Maria is a computer science

student. Therefore, Maria gets less sleep than someone else. $[$C$($x$),$ W$($x$,$ y$),$ S$($x$,$ y$),$ m$]$

$($b$)$ Every member of the board comes from industry or government. Everyone from

government who has a law degree is in favor of the motion. John is not from the industry,

but he does have a law degree. Therefore, if John is a member of the board, he is in favor of

the motion. $M(x),I(x),G(x),L(x),F(x),j$