---
title: "Rules of inference: sailing race"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:rules-inference-sailing-race"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/rules-inference-sailing-race
site: BCT Engineering
---

# Rules of inference: sailing race

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2069 Bhadra, 8 marks
- Video: [Arguments Building Using Rules of Inference (Part 2)](https://www.youtube.com/watch?v=CqOn_n1PoQ4) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/rules-inference-sailing-race
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

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.

## 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.

## Answer

#### 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
| Step | Statement | Reason |
|---|---|---|
| 1 | r → t | Hypothesis 2 |
| 2 | ¬t | Hypothesis 3 |
| 3 | ¬r | Modus tollens from 1, 2 |
| 4 | ¬r ∨ ¬s | Addition 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 |
| 8 | p ∧ q | De Morgan's law and double negation from 7 |
| 9 | p | Simplification from 8 |

So it rained (and it was foggy). ∎

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. 'If it does not rain or if it is not foggy, then the race will be held and the demonstration will go on' is?
   A. (¬p ∧ ¬q) → (r ∧ s)
   B. (¬p ∨ ¬q) → (r ∧ s)
   C. (¬p ∨ ¬q) → (r ∨ s)
   D. (p ∨ q) → (r ∧ s)
   Answer: B. 'or' in the antecedent and 'and' in the consequent give (¬p ∨ ¬q) → (r ∧ s). (Key point: Symbolising)

2. From r → t and ¬t what do we get?
   A. r
   B. ¬r by modus tollens
   C. t
   D. ¬s
   Answer: B. Modus tollens gives ¬r. (Key point: First derivation)

3. ¬r ∨ ¬s is converted to ¬(r ∧ s) using which law?
   A. Distributive law
   B. De Morgan's law
   C. Absorption
   D. Contrapositive
   Answer: B. De Morgan: ¬r ∨ ¬s ≡ ¬(r ∧ s). (Key point: Not both)

4. After ¬(¬p ∨ ¬q) we get p ∧ q. What does this show?
   A. It did not rain
   B. It rained and it was foggy
   C. It was foggy but did not rain
   D. The race was held
   Answer: B. p ∧ q means it rained and it was foggy, so in particular it rained. (Key point: Final result)

## Related in Discrete Structure

- [Rules of inference: Randy and the job](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-randy-job) — Discrete Structure, chapter 1
- [Rules of inference: cheque book argument](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-cheque-book) — Discrete Structure, chapter 1
- [Rules of inference: e-mail and program](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-email-program) — Discrete Structure, chapter 1
- [Rules of inference: movie and sunset](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-rainy-movie) — Discrete Structure, chapter 1
- [Rules of inference: Sunday test in MFC](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-sunday-test) — Discrete Structure, chapter 1
- [Resolution: Tuesday test argument](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Aresolution-tuesday-discrete-structure) — Discrete Structure, chapter 1

## Related in other subjects

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- [Why FOPL; rules of inference](https://www.bctengineer.com/question.md?key=semester_6%2Fartificial_intelligence%3Afopl-rules-of-inference-learning) — Artificial Intelligence, chapter 4