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

# Rules of inference: cheque book argument

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2074 Bhadra, 8 marks
- Video: [Arguments Building Using Rules of Inference (Part 1)](https://www.youtube.com/watch?v=28lebQ60TCc) — 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-cheque-book
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Use rules of inference to show that the hypothesis "If my cheque book is in office, then I have paid my telephone bill", "I was looking for phone bill at breakfast or I was looking for phone bill in my office", "If I was looking for phone bill at breakfast then my cheque book is on breakfast table", "If I was looking for phone bill in my office then my cheque book is in my office", "I have not paid my phone bill" imply the conclusion "My cheque book is on my breakfast table."

## Short answer

p→q and ¬q give ¬p (modus tollens). s→p with ¬p gives ¬s (modus tollens). r∨s and ¬s yield r (disjunctive syllogism). r→t and r give t (modus ponens). So the cheque book is on the breakfast table.

## Answer

#### Propositions
- p: my cheque book is in the office
- q: I have paid my telephone bill
- r: I was looking for the phone bill at breakfast
- s: I was looking for the phone bill in my office
- t: my cheque book is on the breakfast table

#### Hypotheses and conclusion
1. p → q
2. r ∨ s
3. r → t
4. s → p
5. ¬q

Conclusion: t.

#### Proof
| Step | Statement | Reason |
|---|---|---|
| 1 | p → q | Hypothesis |
| 2 | ¬q | Hypothesis |
| 3 | ¬p | Modus tollens from 1, 2 |
| 4 | s → p | Hypothesis |
| 5 | ¬s | Modus tollens from 4, 3 |
| 6 | r ∨ s | Hypothesis |
| 7 | r | Disjunctive syllogism from 6, 5 |
| 8 | r → t | Hypothesis |
| 9 | t | Modus ponens from 8, 7 |

So the cheque book is on the breakfast table. ∎

I also checked all 32 truth assignments by computer: every assignment that makes the five hypotheses true makes t true, so the argument is valid.

## Check yourself

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

1. The statement 'I was looking at breakfast of I was looking in my office' is read as which?
   A. r ∧ s
   B. r → s
   C. r ∨ s
   D. r ↔ s
   Answer: C. 'of' is a typo for 'or', so the hypothesis is r ∨ s. (Key point: Translating 'or')

2. From p → q and ¬q, what do we get first?
   A. q
   B. ¬p by modus tollens
   C. p by modus ponens
   D. r
   Answer: B. Modus tollens: from p → q and ¬q, conclude ¬p. (Key point: First step)

3. How do we derive ¬s?
   A. Modus ponens from s → p and p
   B. Modus tollens from s → p and ¬p
   C. Addition
   D. Simplification from r ∨ s
   Answer: B. s → p with ¬p gives ¬s by modus tollens. (Key point: Eliminating s)

4. Which rule turns r ∨ s and ¬s into r?
   A. Hypothetical syllogism
   B. Disjunctive syllogism
   C. Resolution of p
   D. Conjunction
   Answer: B. Disjunctive syllogism: from a ∨ b and ¬b, conclude a. (Key point: Reaching r)

## 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: sailing race](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-sailing-race) — 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
- [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: e-mail and program](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Arules-inference-email-program) — 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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