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.”

Consider 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 om

The other hypotheses as follows

If the sailing race is held, then the trophy will be awarded.

The trophy was not awarded imply the conclusion is "It rained".

The objective is to use the rules of inference to show the above hypotheses. 

Assume that the propositions are,

Let r be the proposition "It rains", let f be the proposition "It is foggy", let s be the proposition

"The sailing race will be held", let l be the proposition "The lifesaving demonstration will go on",

and let t be the proposition 'The trophy Will be awarded 

It does not rain is represented as $\neg r$ , it is not foggy is represented as $\neg f$ .

The trophy was not awarded is represented as $\neg t$ .

The premises written as

$(\neg r \vee \neg f) \rightarrow(s \wedge l), s \rightarrow t \ and \ \neg t$

Conclude part is r.

Set up the proof in two columns, with reasons.

Note that it is valid to replace subexpressions by other expressions logically equivalent to them.