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Rules of inference: e-mail and program

Short answer

Let p: send email, q: finish program, r: sleep early, s: refreshed. p→q contrapositive ¬q→¬p. ¬p→r, so ¬q→r by hypothetical syllogism. r→s gives ¬q→s. Thus if I don't finish the program, I will wake up feeling refreshed.

Using rules of inferences, show that the hypotheses "If you send me an e-mail message, then I will finish writing the program," "If you do not send me an e-mail message, then I will go to sleep early," and "If I go to sleep early, then I will wake up feeling refreshed" lead to the conclusion "If I do not finish writing the program, then I will wake up feeling refreshed." You are required to show each steps and give reasons for those steps before you come to the desired conclusion from the hypotheses.

Propositions

  • p: you send me an e-mail message
  • q: I will finish writing the program
  • r: I will go to sleep early
  • s: I will wake up feeling refreshed

Hypotheses and conclusion

  1. p → q
  2. ¬p → r
  3. r → s

Conclusion: ¬q → s.

Proof

StepStatementReason
1p → qHypothesis 1
2¬q → ¬pContrapositive of 1 (logical equivalence)
3¬p → rHypothesis 2
4¬q → rHypothetical syllogism from 2, 3
5r → sHypothesis 3
6¬q → sHypothetical syllogism from 4, 5

So, if I do not finish writing the program, then I will wake up feeling refreshed. ∎

✓ Answer checked by Claude Opus 5.5 (high effort)

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