Properties of NOR

Theorem
Let $$\downarrow$$ signify the NOR operation.

The following results hold.

NOR with itself is the Not operation:


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

NOR is commutative:


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

NOR is not associative:


 * $$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$$

Proof by Natural deduction
Commutativity is proved by the Tableau method:

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

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

As can be seen, $$v \left({p \downarrow \left({q \downarrow r}\right)}\right) \ne v \left({\left({p \downarrow q}\right) \downarrow r}\right)$$ except in four combinations of values, that is, when $$v \left({p}\right) = v \left({r}\right)$$.

So $$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$$ nor the other way about.