- 2073 Magh · 8 marks
Rules of inference: Sunday test in MFC
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.
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.
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
- p → (q ∧ r)
- s → ¬r
- p
- 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.
✓ Answer checked by Claude Opus 5.5 (high effort)
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