How to proof "(∧)→¬(¬∨¬)" is a tautology

Question

That is part of my homework and I really have no idea about that. Really need some help.

There is the question:

In our proof system for Sentential, if there is a proof of ℬ from then →ℬ is a tautology. Use a proof to show that the following sentence is a tautology. (∧)→¬(¬∨¬)

Answer 1

The easiest way to go about proving that something involving only a few number of independent statements is a tautology, is to make use of the truth table. The underlying principle which you are appealing to is the completeness theorem, which is well known.

So, start doing like this:

• Form a truth table, where you have one column for the truth values of $P$, and one column for the truth value of $Q$.
• Then find out what are the various truth values for the two things in question you get for various combinations of the truth values of $P$ and $Q$.
• Observe whether these two truth values always match or not. If they do then you have a tautology, and otherwise you don't.

Answer 2

Oh, I took this yesterday in my discrete mathematics class. you could either use a truth table, or just use the double negation and distributive laws. Basically $\neg(\neg \vee \neg )$ becomes $(\wedge)$ because $\neg(\neg P)= P$, $\neg (\neg Q)$ and negating the or operator makes it an and operator(because logic!). So the end result is $(\wedge )→(\wedge )$.