Functionally Complete Logical Connective

Question

I am trying to check if a new logical connective is functionally complete, or if a combination of it with "not" is functionally complete. The connective is: not(p or q). In addition I am trying to find a CNF and DNF forms of it, which I think I know how to do. Regarding the completeness, what exactly should I look for? Thank you.

Answer 1

Yes it is complete, it's called the $NOR$ (a contraction of 'not .. or ..' of course!)

To show it is (by itself!) complete, you need to demonstrate it can be used to express any truth-function.

Now, a typical starting point is to begin with a set of operators we already know to be complete, e.g. $\{ \land, \lor, \neg \}$

Then, because of DeMorgan and Double Negation, we can rewrite any $\lor$ in terms of $\land$ and $\neg$, so $\{ \land , \neg \}$ is complete as well.

Finally, show that $\land$ and $\neg$ can be expressed in terms of $NOR$:

$P \: NOR \: P \Leftrightarrow \neg (P \lor P) \Leftrightarrow \neg P$

So:

$(P \: NOR \: P) \: NOR \: (Q \: NOR \: Q) \Leftrightarrow \neg((P \: NOR \: P) \lor (Q \: NOR \: Q)) \Leftrightarrow \neg(\neg P \lor \neg Q) \Leftrightarrow P \land Q$