. Check the validity of the following arguments (a) Hayder works hard. If Hayder works hard, then he is a dull boy. If Hayder is a dull boy, then he will not get the job. therefore, Randy will not get the job. (b) If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on. If the sailing race is held, then the trophy will be awarded. The trophy was not awarded. Thus, it rained.

Let us check the validity of the following arguments.

(a) Hayder works hard. If Hayder works hard, then he is a dull boy. If Hayder is a dull boy, then he will not get the job. Therefore, Randy will not get the job.

Denote by $p$ the statement "Hayder works hard", by $q$ the statement "He is a dull boy", by $r$ the statement "He will not get the job". Then the premises are $p, p\to q,$ and $q\to r.$ According to the rule Modus ponens, we get from $p, p\to q$ the statemet $q.$ In the same way using Modus ponens to $q,q\to r,$ we have the conclusion $r.$

(b) If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on. If the sailing race is held, then the trophy will be awarded. The trophy was not awarded. Thus, it rained.

Denote by $p$ the statement "It rained", by $q$ the statement "it is foggy", by $r$ the statement "The sailing race will be held" and by $s$ the statement "The lifesaving demonstration will go on", by $t$ the statement "The trophy will be awarded". Then the premises are $\neg p\lor\neg q\to r\land s, r\to t,$ and $\neg t.$

According to the Modus tollens rule, we get from $\neg t, r\to t$ the statement $\neg r.$

Using Intoduction disjunction rule, we have $\neg r\lor\neg s.$ According to de Morgan's laws, we have $\neg(r\land s).$ According to the rule Modus tollens, we get from $\neg(r\land s), \neg p\lor\neg q\to r\land s$ the statemet $\neg(\neg p\lor\neg q)=p\land q.$ And using the conjunction exclusion rule, we have the conclusion $p.$