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

# Rules of inference: movie and sunset

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Bhadra, 8 marks
- Video: [RULES of INFERENCE - DISCRETE MATHEMATICS](https://www.youtube.com/watch?v=8DW0K3mnc-0) — TrevTutor
- 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-rainy-movie
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Using rules of inferences, show that the hypothesis "It is not rainy today and its hotter than yesterday", "We will go for movie only if it is rainy", "If we do not go for movie, then we will go for shopping", and "If we go for shopping, then we will be home by sunset" lead to the conclusion "We will be home by sunset". You are required to show each steps and give reasons for those steps before you come to desired conclusion from the hypothesis.

## Short answer

Let p: rainy, q: hotter, m: movie, s: shopping, h: home by sunset. From ¬p ∧ q, simplify ¬p. m → p (only if) with ¬p gives ¬m by modus tollens. ¬m → s gives s by modus ponens. s → h gives h by modus ponens. So we will be home by sunset.

## Answer

#### Propositions
- p: it is rainy today
- q: it is hotter than yesterday
- m: we go for a movie
- s: we go for shopping
- h: we will be home by sunset

#### Hypotheses and conclusion
1. ¬p ∧ q
2. m → p  ("m only if p")
3. ¬m → s
4. s → h

Conclusion: h.

#### Proof
| Step | Statement | Reason |
|---|---|---|
| 1 | ¬p ∧ q | Hypothesis 1 |
| 2 | ¬p | Simplification from 1 |
| 3 | m → p | Hypothesis 2 |
| 4 | ¬m | Modus tollens from 3, 2 |
| 5 | ¬m → s | Hypothesis 3 |
| 6 | s | Modus ponens from 5, 4 |
| 7 | s → h | Hypothesis 4 |
| 8 | h | Modus ponens from 7, 6 |

Hence we will be home by sunset. ∎

The phrase "A only if B" is translated as A → B, which is why hypothesis 2 is m → p and not p → m.

## Check yourself

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

1. 'We will go for a movie only if it is rainy' is symbolised as?
   A. p → m
   B. m → p
   C. m ↔ p
   D. ¬m → ¬p
   Answer: B. 'A only if B' means A → B, so m → p. (Key point: 'Only if')

2. Which rule gives ¬m from m → p and ¬p?
   A. Modus ponens
   B. Modus tollens
   C. Disjunctive syllogism
   D. Addition
   Answer: B. Modus tollens: from a → b and ¬b conclude ¬a. (Key point: Deriving ¬m)

3. Which step then gives s (go shopping)?
   A. Modus tollens with ¬m → s
   B. Modus ponens with ¬m → s and ¬m
   C. Simplification of ¬p ∧ q
   D. Hypothetical syllogism
   Answer: B. From ¬m → s and ¬m, modus ponens gives s. (Key point: Using ¬m)

4. How is ¬p obtained from ¬p ∧ q?
   A. Addition
   B. Simplification
   C. Conjunction
   D. Modus ponens
   Answer: B. Simplification extracts one conjunct. (Key point: Getting ¬p)

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