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I'm new to discrete mathematics and I have the following assignment prompt.

This week's proof will be in two parts. First, you will prove that the set of logical operators {AND, OR, NOT} is functionally complete.

That is, you'll prove that ALL 16 binary logical operators can be written in terms of only these three. Secondly, you'll prove that each of AND, OR, NOT can be written in terms of NAND.

Conclude by explaining why this shows that NAND is functionally complete. Good luck!!!

I've been able to do most of this just fine but two operators in particular are tripping me up; ~P and ~Q.

The examples given lead me to believe the instructor wants us to express ¬ in terms of ∨ and ∧.

What the above statement is saying is that ANY logical operator can be expressed using only the functionally complete operator. Some examples will be helpful here. Let's say that you wanted to express the AND and IMPLICATION operators in terms of only OR and NOT

P AND Q = ~(~P OR ~Q)

P -> Q = ~P OR Q

Since the truth tables for ∨ and ∧ guarantee that when P and Q are both True they will evaluate to True I'm not sure what I can do to show ~P. It seems silly to write ~P = ~P or ~Q = ~Q so I figure I must be misinterpreting something. Could someone please point out where I am mistaken?

I'm showing the 16 operators from the table here.

P.S. I can easily show ∨ and ∧ using DeMorgan's Law, but I don't know how I could express ~P without using ~.

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    What is P NAND P? – 2017-01-21
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    @JohnB It's obviously ¬P. I didn't think I would be able to use ¬ as part of a ¬P definition. I suppose you're correct. – 2017-01-21

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You aren't being asked to express $\neg$ in terms of $\wedge$ and $\vee$: as you have correctly observed, that's not going to be possible. Instead, you are being asked to express $\neg$ (and all other logical operators) in terms of NAND. The examples are just examples of what "express some operation in terms of some other operations" means.

When expressing operations in terms of operations, there is also nothing wrong with doing so "trivially". So if you want to express $\neg P$ using $\neg$, $\wedge$, and $\vee$, it is perfectly correct to just do so by writing $\neg P$.

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    To do that the instructions want me to (1) Define all 16 in terms of "{AND, OR, NOT}". So my first task is to do that. (2) show those in terms of NAND, as you said. Since (1) requires the use of those three would ¬P = ¬P? It seems redundant to do so. – 2017-01-21
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    Oh, I see what you mean. Then yes, it is fine to just say $\neg P=\neg P$. That writes the operation $\neg P$ in terms of $\neg$, $\wedge$, and $\vee$. – 2017-01-21