[Math] How to prove that a statement is a tautology using logical equivalences

Question

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:

(𝑝∧𝑞) → (𝑝∨𝑞)

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. (𝑝∧𝑞) → (𝑝∨𝑞)
2. ~(𝑝∧𝑞) ∨ (𝑝∨𝑞)
3. (~𝑝∨~𝑞) ∨ (𝑝∨𝑞)

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

Answer 1

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.