---
title: "Rules of inference: Sunday test in MFC"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:rules-inference-sunday-test"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/rules-inference-sunday-test
site: BCT Engineering
---

# Rules of inference: Sunday test in MFC

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2073 Magh, 8 marks
- Video: [Btech Discrete maths/MFCS |Inference theory|premises |consistency of premises|mathematical logic](https://www.youtube.com/watch?v=ijW7-wdXoU8) — RS 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-sunday-test
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Hypothesis: "If today is Sunday then I will have a test in MFC and IT. If my IT teacher is sick then I will not have a test in IT. Today is Sunday and my IT teacher is sick." Conclusion: "I will have a test in MFC." Use rule of inference to prove it.

## Short answer

From p (Sunday) and p→(q∧r), modus ponens gives q∧r. Simplification yields q, the test in MFC. The conclusion follows directly; the premises about the IT teacher being sick are not needed for this deduction.

## Answer

#### Propositions
- p: today is Sunday
- q: I will have a test in MFC
- r: I will have a test in IT
- s: my IT teacher is sick

#### Hypotheses and conclusion
1. p → (q ∧ r)
2. s → ¬r
3. p
4. s

Conclusion: q.

#### Proof
| Step | Statement | Reason |
|---|---|---|
| 1 | p | Hypothesis |
| 2 | p → (q ∧ r) | Hypothesis |
| 3 | q ∧ r | Modus ponens from 1, 2 |
| 4 | q | Simplification from 3 |

So I will have a test in MFC. ∎

**Remark.** Steps 1–4 already prove the conclusion. The other hypotheses are not needed. Indeed, with s true and s → ¬r we would get ¬r, while step 3 gives r, so the hypotheses together are inconsistent. A valid argument from inconsistent hypotheses is still valid, and the proof above does not depend on the inconsistency.

## Check yourself

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

1. Which two lines give q ∧ r?
   A. p and s → ¬r
   B. p and p → (q ∧ r)
   C. s and s → ¬r
   D. q and r
   Answer: B. Modus ponens on p and p → (q ∧ r) gives q ∧ r. (Key point: Modus ponens)

2. Which rule gives q (the MFC test) from q ∧ r?
   A. Addition
   B. Simplification
   C. Modus tollens
   D. Disjunctive syllogism
   Answer: B. Simplification: from a ∧ b conclude a. (Key point: Simplification)

3. Are the hypotheses about the sick teacher needed to prove q?
   A. Yes, otherwise r is unknown
   B. No, steps from p and p → (q ∧ r) already prove q
   C. Yes, to use modus tollens
   D. Yes, because they make the premises consistent
   Answer: B. The proof uses only hypotheses 1 and 3; the others are unnecessary. (Key point: Unneeded hypotheses)

4. The hypotheses give both r and ¬r. What follows about the argument?
   A. The argument is invalid
   B. The argument is still valid since the proof used valid rules
   C. The conclusion must be false
   D. No conclusion can be drawn
   Answer: B. A valid derivation stays valid even when the hypotheses are inconsistent. (Key point: Inconsistent hypotheses)

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