The $\text{NOR}$ operation is defined at follows $x \text{NOR} y = \text{NOT}(x+y)$. How do I prove that $\{\text{NOR}\}$ is functionally complete?

Question

The $\text{NOR}$ operation is defined at follows $x\;\text{NOR}\;y = \text{NOT}(x+y)$. How do I prove that $\{\text{NOR}\}$ is functionally complete?

I need help solving it as I don't know how to.

Answer 1

Let's denote NOR with Pierce's arrow $\downarrow$ so$$x\downarrow x=\neg x,\,\neg(x\downarrow y)=x+y,(\neg x)\downarrow(\neg y)=x\land y.$$