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

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.

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

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.

Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-307-qpid-194