$($Answer must be in your own hand$-$writing. Affix your signature

in each page of your answer.$)$

$($a$)$ Show that $(pr)vv(qr)$ and $(p^{{?}^{?}}q)r$ are logically equivalent.

$\%$

$(2$ marks$)$

$($b$)$ Show that the premises "A passenger in this compartment has not bought

the travel ticket," and "Everyone in this compartment is seated" imply the

conclusion "Someone in this compartment who has a seat has not bought the

travel ticket."

$(2$ marks$)$

$\%$

$($c$)$ Prove by Mathematical Induction that $n^3+(n+1{)}^3+(n+2{)}^3$ is divisible

by $9$ for all ninN.

$\%$

$(2$ marks$)$