Rule of Association

Definition
This rule is two-fold:


 * Conjunction is associative:
 * $$p \and \left({q \and r}\right) \dashv \vdash \left({p \and q}\right) \and r$$


 * Disjunction is associative:
 * $$p \or \left({q \or r}\right) \dashv \vdash \left({p \or q}\right) \or r$$

Its abbreviation in a tableau proof is $$\textrm{Assoc}$$.

Proof by Natural Deduction
These are proved by the Tableau method.

$$\left({p \and q}\right) \and r \vdash p \and \left({q \and r}\right)$$

is proved similarly.

$$\left(p \or q\right) \or r \vdash p \or \left(q \or r\right)$$

is proved similarly.

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