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:

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

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.