How can I express xNORy solely with NAND operations?

Question

I've tried every which way I can think of to manipulate the algebra using the various laws I was given, but I cannot figure out a way to get $\overline{x+y}$ to convert to only NAND operations using $\overline{x\cdot x}$, $\overline{x\cdot y}$, or $\overline{y\cdot y}$

I've tried using idempotent, DeMorgan's, identity, and double negation laws to no avail.

Answer 1

I’ll use $\mid$ for the connective NAND. You want $\overline{x+y}$; as a first step, this is $(x+y)\mid(x+y)$. Now check that $x+y=(x\mid x)\mid(y\mid y)$, and put the pieces together.