Question

Use rules of inference to show that the hypotheses "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," and "The trophy was not awarded" imply the conclusion
"It rained."

Short answer

The rule of interference used to conclude some hypotheses can be determined.

Step-by-step solution

Find the first premise using De Morgan

Let’s take p be the proposition “It rained” and q be the “It is foggy” and r be the “sailing race is held” and s be “the lifesaving demonstration will go on” and w be “the trophy is awarded”. The form of argument will be,

The first premises using De Morgan’s can be simplified as,

\(\begin{aligned}{l}(\neg p\, \vee \,\neg q) \to (r \wedge s) &= \neg (p \wedge q) \to (r \wedge s)\\ = (p \wedge q) \vee (r \wedge s)\\ = p \wedge s\end{aligned}\)

Find the second and third premises

The second and third premises can imply using disjunctive syllogism rule. Then,

\(\begin{aligned}{l}p \wedge s\\\neg s\\\therefore p\end{aligned}\)

It is concluded that it rained.