Functionally Complete Logical Connectives/NAND

Theorem
The singleton set containing the following logical connective:
 * $\left\{{\uparrow}\right\}$: NAND

is functionally complete.

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

From NAND with Equal Arguments:
 * $\neg p \dashv \vdash p \uparrow p$

From Conjunction in terms of NAND:
 * $p \land q \dashv \vdash \left({p \uparrow q}\right) \uparrow \left({p \uparrow q}\right)$

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, $\left\{{\uparrow}\right\}$ is functionally complete.

Also see

 * NOR is Functionally Complete