$1.($a$)$ Translate each of the following English sentences into the syntax of propositional logic using the

indicated letters. For each letter, indicate explicitly the statement it stands for.

i$.$ Earning a final mark of $70\%$ in the computational logic course is a necessary condition for a

student to obtain their master$$s degree. $(,$ p$)$

ii$.$ If computational logic is a difficult subject, then Bobby and Connie will pass only if they study.

$($d$,$ b$,$ c$,$ s$)$

iii. If Q is a quadrilateral, then Q is a parallelogram if and only if its opposite sides are both equal

and parallel. $($q$,$ p$,$ e$,)$

$($b$)$ Translate the following argument from Plato$$s Crito into the syntax of propositional logic, and then

prove that the argument is valid.

Premise $1.$ If Socrates did not approve of the laws of Athens, then he would either have left Athens

or else have tried to change the laws.

Premise $2.$ If Socrates did not leave Athens and did not try to change the laws, then he agreed

to obey the laws.

Premise $3.$ Socrates did not leave Athens.

Conclusion. If Socrates did not try to change the laws, then he approved of the laws and agreed

to obey them.

Hint. You can prove validity by contradiction: assume that there exists an interpretation where all

premises are true but the conclusion is false.