Using Rules of Inference to imply conclusion This is Discrete Math.  I just learned how to use the rules of inference and I'm not sure if im doing this right at all.  The question is:

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."
Here is an outline.

"Trophy not awarded" and "If sailing race held, then trophy will be awarded" $\implies$ $A$

(What is $A$?)

$A$ and "If it does not rain or if it is not foggy, then the sailing race will be held" $\implies $B

(What is $B$?)
Let $p$ be the proposition "it rain", $q$ be "it is foggy", $r$ be "sailing race is held" , $s$ be "the lifesaving demonstration go on" and $w$ be "the trophy is awarded" .

Then the argument form will be:
\begin{align*}
&amp;(\neg p\vee \neg q)\to (r\wedge s) \\
&amp;s\to w\\
&amp;\neg w\\
&amp;\hline{\therefore\neg p}\\
&amp;
\end{align*}
Now the first premises can be simplified using  De Morgan's, implication and simplification i.e.
\begin{equation*}
\begin{split}
(\neg p\vee \neg q)\to (r\wedge s) &amp; \equiv \neg (p\wedge q)\to (r\wedge s) \quad \text{by De Morgan's rule}\\
&amp; \equiv (p\wedge q)\vee (r\wedge s) \quad \text{by De Implication rule}\\
&amp;  \equiv p\vee s \quad \text{by Simplification rule}
\end{split}
\end{equation*}
The second and third premises imply $\neg s$ by $\textit{disjunctive syllogism}$ rule. Then we have
\begin{align*}
&amp;p\vee s \\
&amp;\neg s\\
&amp;\hline{\therefore\neg p}\\
&amp;
\end{align*}
The conclusion also follows by disjunctive syllogism.