Solution to Given the following 2 premises, 1. 𝑝 → (𝑞 ∨ 𝑟) 2. 𝑞 → 𝑠 … - Sikademy
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Archangel Macsika

Given the following 2 premises, 1. 𝑝 → (𝑞 ∨ 𝑟) 2. 𝑞 → 𝑠 Prove 𝑝 → (𝑟 ∨ 𝑠) is valid using the Proof by Contradiction method.

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Solution:

Proof by Contradiction Method:


  1. p\rightarrow (q\lor r) Premise
  2. q\rightarrow s Premise
  3. \neg (p\rightarrow (r\lor s)) Premise, proof by contradiction
  4. \neg (\neg p\lor (r\lor s)) 3, Definition of \rightarrow
  5. p\land \neg (r\lor s) 4, DeMorgan’s law
  6. p 5, Specialization
  7. \neg (r\lor s) 5, Specialization
  8. \neg r\land \neg s 7, DeMorgan’s law
  9. \neg r 8, Specialization
  10. \neg s 8, Specialization
  11. \neg q 2, 10, Modus Tollens
  12. \neg q\land \neg r 9, 11
  13. \neg (q\lor r) 12, DeMorgan’s law
  14. \neg p 1, 13, Modus Tollens
  15. False 6, 14, proof by contradiction


Premise \neg (p\rightarrow (r\lor s)) was false, so p\rightarrow (r\lor s) must be true.

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Question ID: mtid-5-stid-8-sqid-306-qpid-193