Skip to main content

Rules of inference: sailing race

Short answer

Let p: rain, q: fog, r: race, s: lifesaving, t: trophy. r→t and ¬t give ¬r by modus tollens. ¬r implies ¬r ∨ ¬s, i.e., ¬(r ∧ s). (¬p ∨ ¬q)→(r ∧ s) and ¬(r ∧ s) give ¬(¬p ∨ ¬q) by modus tollens, which is p ∧ q; hence p: it rained.

Construct an argument using 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." You are required to show each step and give reasons for those steps before you come to the desired conclusion from the hypotheses.

Propositions

  • p: it rains
  • q: it is foggy
  • r: the sailing race will be held
  • s: the lifesaving demonstration will go on
  • t: the trophy will be awarded

Hypotheses and conclusion

  1. (¬p ∨ ¬q) → (r ∧ s)
  2. r → t
  3. ¬t

Conclusion: p ("It rained").

Proof

StepStatementReason
1r → tHypothesis 2
2¬tHypothesis 3
3¬rModus tollens from 1, 2
4¬r ∨ ¬sAddition from 3
5¬(r ∧ s)De Morgan's law from 4
6(¬p ∨ ¬q) → (r ∧ s)Hypothesis 1
7¬(¬p ∨ ¬q)Modus tollens from 6, 5
8p ∧ qDe Morgan's law and double negation from 7
9pSimplification from 8

So it rained (and it was foggy). ∎

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Discrete Structure chapter 1 questions →