---
title: "Rules of inference: Randy and the job"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:rules-inference-randy-job"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/rules-inference-randy-job
site: BCT Engineering
---

# Rules of inference: Randy and the job

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Magh, 8 marks
- Video: [Exercise # 1.6  Q5 to Q8 ( Rules of Inference)|| Rosen Discrete Mathematics 7th Edition|| M.Owais](https://www.youtube.com/watch?v=5dLqVgVX3BM) — Easy Way Mathematics
- 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-randy-job
- 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 "Randy works hard," "If Randy works hard, then he is a dull boy," and "If Randy is a dull boy, then he will not get the job" imply the conclusion "Randy will not get the job."

## Short answer

p: Randy works hard, q: dull boy, r: gets job. From p and p→q, q by modus ponens. q→¬r and q give ¬r by modus ponens. Therefore Randy will not get the job.

## Answer

#### Propositions
- p: Randy works hard
- q: Randy is a dull boy
- r: Randy will get the job

#### Hypotheses and conclusion
1. p
2. p → q
3. q → ¬r

Conclusion: ¬r.

#### Proof
| Step | Statement | Reason |
|---|---|---|
| 1 | p | Hypothesis |
| 2 | p → q | Hypothesis |
| 3 | q | Modus ponens from 1, 2 |
| 4 | q → ¬r | Hypothesis |
| 5 | ¬r | Modus ponens from 3, 4 |

So Randy will not get the job. ∎

(Alternatively: p → q and q → ¬r give p → ¬r by hypothetical syllogism, and then modus ponens with p gives ¬r.)

## Check yourself

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

1. From p and p → q, which rule gives q?
   A. Modus tollens
   B. Modus ponens
   C. Addition
   D. Resolution on q
   Answer: B. Modus ponens. (Key point: First inference)

2. How is ¬r reached after getting q?
   A. Modus tollens with q → ¬r
   B. Modus ponens with q → ¬r
   C. Simplification
   D. Disjunctive syllogism
   Answer: B. q and q → ¬r give ¬r by modus ponens. (Key point: Final step)

3. Which alternative derivation of p → ¬r is mentioned?
   A. Modus tollens
   B. Hypothetical syllogism on p → q and q → ¬r
   C. Disjunctive syllogism
   D. Addition
   Answer: B. Chaining p → q and q → ¬r by hypothetical syllogism gives p → ¬r. (Key point: Alternative proof)

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## Related in other subjects

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