NOR is Commutative

Theorem

Let $\downarrow$ signify the NOR operation.

Then, for any two propositions $p$ and $q$:

$p \downarrow q \dashv \vdash q \downarrow p$

That is, NOR is commutative.

Proof by Tableau

Proceed by the tableau method:

$p \downarrow q \vdash q \downarrow p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \downarrow q$	$\mathrm P$	(None)
2		1	$\neg \left({p \lor q}\right)$	By definition	1
4		1	$\neg \left({q \lor p}\right)$	$\mathrm {Comm}$	1
5		1	$q \downarrow p$	By definition	1

$q \downarrow p \vdash p \downarrow q$ is proved similarly.

$\blacksquare$

Proof by Truth Table

Apply the Method of Truth Tables:

$\begin{array}{|ccc||ccc|} \hline p & \downarrow & q & q & \downarrow & p \\ \hline F & T & F & F & T & F \\ F & F & T & T & F & F \\ T & F & F & F & F & T \\ T & F & T & T & F & T \\ \hline \end{array}$

As can be seen by inspection, the truth values under the main connectives match for all models.

$\blacksquare$