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Prove that (p → q) → ((r ∨ p) → (r ∨ q)) is a tautology without using truth table


I can't find a proper formula for this considering it's almost exclusively made up of implications. Can somebody help me?

just I want to know how to simplify it.

I tried this but still no correct answerenter image description here


Solution

  • If r is false, then ((r ∨ p) → (r ∨ q)) is just (p → q).

    If r is true, then ((r ∨ p) → (r ∨ q)) is just (true → true).