$4.($Total points $=20)$ Use the rules of inference and the laws of propositional logic to prove that

each of the following arguments are valid.

$($a$)$

p $->$ q

q $->$ r

$$r

$-------------$

$$p

Page $3$ of $3$

$($b$)$

p $->($q $$ r$)$

$$q

$-------------$

$$p

$($c$)$

p $$ q

$$p $$ r

$$q

$---------------$

$$ r

$($d$)$

p $$ q

q $->$ r

p $$ s $->$ t

$$r

$$q $->$ u $$ s

$---------------$

$$ t