# Show that a set of logical connectives given with W(p, q, r) is complete

I just stuck upon a discrete maths-related problem with proving that a set of logical connectives is complete. I already know how to prove that ("basic") sets are full (examples like show that the given set of connectives is complete: $$\{\lnot, \Rightarrow\}$$), but I got stuck upon the following example...

Three-figure logical connective $$W(p,q,r)$$ is defined as $$W(p,q,r) \equiv (p \lor q) \Rightarrow \lnot r$$. Proove that set $$\{W\}$$ is a complete set of connectives.

What should I do here? I know that a known complete set is $$\{\lnot, \land, \lor\}$$ and that I should "derive" each one of those connectives with $$(p \lor q) \Rightarrow \lnot r$$, but how should I even start?

P.S.: I'm sorry if the translation of the assignment isn't 100% accurate, because English is not my first language and this assignment is given in my native language :)

• $p \lor q \Rightarrow \lnot r$ is equivalent to $(\lnot p \land \lnot q)\lor \lnot r$ Aug 24, 2019 at 11:04

W(p,p,p) is equivalent to $$p \implies \neg p$$
which is equivalent to $$\neg p \lor \neg p.$$
$$W(p,p,\neg q)$$
is equivalent to $$p \implies q.$$
Exercise. Show {$$p \implies \neg q$$} is a complete set of connectives.
Hint: see what happens when you use only one or two distinct arguments. For example, what does $$W(p, p, p)$$ yield?