Answer to Question #294522 in Discrete Mathematics for atul

Question

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.”

Answer 1

Using the premises ”If the sailing race is held, then the trophy will be awarded,” and ”The trophy was not awarded”, we conclude by Modus Tollens Rule the proposition "The sailing race will not be held".

It follows from the proposition "The sailing race will not be held" and the Rule of Introduction of Disjunction the proposition "The sailing race will not be held or the lifesaving demonstration will not go on".

De Morgan's Law implies from proposition "The sailing race will not be held or the lifesaving demonstration will not go on" the proposition "It is not true that the sailing race will be held and the lifesaving demonstration will go on".

Using Modus Tollens to the premise "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” and to the statement "It is not true that the sailing race will be held and the lifesaving demonstration will go on", we conclude the proposition "It is not true that it does not rain or it is not foggy".

De Morgan's Law implies from proposition "It is not true that it does not rain or if it is not foggy" the proposition "It rained and it is foggy".

Using Exclusion Conjunction Rule to the proposition "It rained and if it is foggy" we get the conclusion "It rained".