Functionally Complete Logical Connectives/NOR

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Theorem

The singleton set containing the following logical connective:

$\set {\downarrow}$: NOR

is functionally complete.


Proof

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


From NOR with Equal Arguments:

$\neg p \dashv \vdash p \downarrow p$


From Disjunction in terms of NOR:

$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \downarrow q}$

demonstrating that $p \lor q$ can be represented solely in terms of $\downarrow$.


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

$\blacksquare$


Also see


Sources