- 2070 Bhadra · 8 marks
Rules of inference: movie and sunset
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.
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.
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
- ¬p ∧ q
- m → p ("m only if p")
- ¬m → s
- 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.
✓ Answer checked by Claude Opus 5.5 (high effort)
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