# Functionally Complete Logical Connectives/NAND

## Theorem

The singleton set containing the following logical connective:

$\set {\uparrow}$: NAND

## Proof

From Functionally Complete Logical Connectives: Negation and Conjunction, any boolean expression can be expressed in terms of $\land$ and $\neg$.

$\neg p \dashv \vdash p \uparrow p$
$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$

demonstrating that $p \land q$ is expressed solely in terms of $\uparrow$.

Thus any boolean expression can be represented solely in terms of $\uparrow$.

That is, $\set {\uparrow}$ is functionally complete.

$\blacksquare$