---
title: "Resolution: Sita and her umbrella"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:resolution-sita-umbrella"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/resolution-sita-umbrella
site: BCT Engineering
---

# Resolution: Sita and her umbrella

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2071 Bhadra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/resolution-sita-umbrella
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Use resolution to show the hypothesis "It is not raining or Sita has her umbrella," "Sita does not have her umbrella or she does not get wet," and "It is raining or Sita does not get wet" imply that "Sita does not get wet."

## Short answer

Let r: raining, u: umbrella, w: wet. Clauses: ¬r ∨ u, ¬u ∨ ¬w, r ∨ ¬w, and w (negation of ¬w). Resolve r ∨ ¬w with w to get r; with ¬r ∨ u yields u; with ¬u ∨ ¬w yields ¬w; resolve with w yields □, proving Sita does not get wet.

## Answer

#### Propositions
- r: it is raining
- u: Sita has her umbrella
- w: Sita gets wet

#### Hypotheses (already clauses)
1. ¬r ∨ u
2. ¬u ∨ ¬w
3. r ∨ ¬w

Conclusion: ¬w. Its negation, **w**, is added as clause 4.

#### Resolution steps
| Step | Resolvent | From | Resolved on |
|---|---|---|---|
| 5 | r | 3, 4 | w |
| 6 | u | 1, 5 | r |
| 7 | ¬w | 2, 6 | u |
| 8 | □ (empty clause) | 4, 7 | w |

The empty clause is obtained, so the hypotheses together with w are unsatisfiable. Hence the hypotheses imply ¬w: **Sita does not get wet.** ∎

## Check yourself

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

1. The conclusion is ¬w (Sita does not get wet). Which clause is added?
   A. ¬w
   B. w
   C. r
   D. ¬r
   Answer: B. We add the negation of the conclusion, which is w. (Key point: Negated conclusion)

2. Resolving r ∨ ¬w with w gives what?
   A. ¬w
   B. r
   C. u
   D. □
   Answer: B. Resolving on w removes ¬w and w, leaving r. (Key point: First resolvent)

3. After r, which clause is resolved with r to get u?
   A. ¬u ∨ ¬w
   B. ¬r ∨ u
   C. r ∨ ¬w
   D. w
   Answer: B. ¬r ∨ u resolved with r on r gives u. (Key point: Chain)

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