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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.

Ѕᴀᴀᴅ
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Nolan D.
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    I suggest writing out the truth tables for "If X then Y" and the four proposed disjunctions "(not X) or (not Y)", "X or Y", "(not X) or Y", and "X or (not Y)". – Andreas Blass Dec 17 '18 at 02:54
  • The very problem is to say that $p\to q$ is the same as $\neg p\vee q$. Usually this is actually the definition. – Matt Samuel Dec 17 '18 at 03:52

2 Answers2

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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".

Ongky Denny Wijaya
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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.

Bram28
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