I'm learning about tautologies right now. I see that a tautology is when two propositional statements have the same truth values. But I'm struggle with the truth table my professor provided about the tautology between "p implies q" and "not p or q"
The first two columns of the truth table are obviously the propositional variables, p and q. The next column represents "not p" which I don't understand why we'd have that considering the first column contains instances of "not p". The fourth column is where I become confused, because from it I get the conclusions, "not p or q" is true when both p and q are true (which I agree with), it is false when p is true and q is false (which I don't understand. I thought as long as at least one propositional variable is true within a disjunction the statement is true as long as its not the exclusive or?, true when p is false and q is true (this makes sense, that's the proposition itself), and true when p is false and q is false (again, I don't understand this. Here we have a disjunction where both variables are false, and yet somehow the proposition "not p or q" is true?)
Can someone explain what I'm missing?