job.”4. Use rules of inference to show that the hypotheses “If it does not rain orif it is not foggy, then the sailing race will be held and the lifesavingdemonstration will go on,” “If the sailing race is held, then the trophywill be awarded,” and “The trophy was not awarded” imply theconclusion “It rained.
Question
job.”4. Use rules of inference to show that the hypotheses “If it does not rain orif it is not foggy, then the sailing race will be held and the lifesavingdemonstration will go on,” “If the sailing race is held, then the trophywill be awarded,” and “The trophy was not awarded” imply theconclusion “It rained.
Solution
The problem can be solved using the rules of inference in propositional logic. Let's denote:
- R: It rains
- F: It is foggy
- S: The sailing race will be held
- L: The lifesaving demonstration will go on
- T: The trophy will be awarded
The hypotheses can be written as:
- ¬R ∨ ¬F → (S ∧ L)
- S → T
- ¬T
We need to prove that these hypotheses imply the conclusion R (It rained).
From hypothesis 2 and 3 (using Modus Tollens), we can infer ¬S (The sailing race was not held).
Now, we know that ¬R ∨ ¬F → (S ∧ L) is equivalent to (R ∧ F) ∨ (S ∧ L) (using the rule of contraposition and De Morgan's laws).
Since we know ¬S, we can infer that R ∧ F must be true (using Disjunctive Syllogism).
Therefore, it rained (R is true).
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