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I'm having trouble understanding how exactly to use laws of logical equivalences to prove what a statement is equivalent to or if it's a tautology. In this particular case, I have the statement:

(∧) → (∨)

which needs to be proven as a tautology. I have all the laws for reference in front of me; I think the next steps would be:

  1. (∧) → (∨)
  2. ~(∧) ∨ (∨)
  3. (~∨~) ∨ (∨)

Following the definition of tautology as being always true, would the end goal statement be p∨T=T?

2 Answers 2

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You are right up to that point. From there you can group $\neg p \vee p $ and $\neg q \vee q$ together, which are true by definition. So you have $True \vee True = True$.

There is your tautology.

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You can use that $P \lor P \Leftrightarrow P$ (twice), then use $P \lor Q \Leftrightarrow Q \lor P$, and finally that $P \lor \neg P \Leftrightarrow \top$.