Why are these two simple statements equivalent?

Question

As I study for my mathematical structures I final, I encountered a problem that I am unable to understand. The problem gives me the statement:

If today is Tuesday, then we have class.

I am being told that the statement that is logically equivalent to this one:

Today is not Tuesday or we have class.

Why is this? I would prefer an explanation using $p→q$ notation, please.

Answer 1

By truth table, the statement $p\implies q$ is equivalent with $\neg p\lor q$.

You can check the statement $(p\implies q)\iff (\neg p\lor q)$ is a tautology statement.

So, "If today is Tuesday, then we have class" equivalent with "Today is not Tuesday or we have class".

Answer 2

The truth-table will show their equivalence ... but if you find the equivalence unintuitive, that's because we typicaly don't regard the English 'if... then ...' as a truth-functional connective, meaning that it doesn't quite match the mathemtcailly defined material implication.