Rule of Association

Definition
This rule is two-fold:


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


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

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

Alternative rendition
These can alternatively be rendered as:


 * $\vdash \left({p \land \left({q \land r}\right)}\right) \iff \left({\left({p \land q}\right) \land r}\right)$
 * $\vdash \left({p \lor \left({q \lor r}\right)}\right) \iff \left({\left({p \lor q}\right) \lor r}\right)$

They can be seen to be logically equivalent to the forms above.

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

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

is proved similarly.

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

is proved similarly.

Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.

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

$$\begin{array}{|ccccc||ccccc|} \hline p & \land & (q & \land & r) & (p & \land & q) & \land & r \\ \hline F & F & F & F & F & F & F & F & F & F \\ F & F & F & F & T & F & F & F & F & T \\ F & F & T & F & F & F & F & T & F & F \\ F & F & T & T & T & F & F & T & F & T \\ T & F & F & F & F & T & F & F & F & F \\ T & F & F & F & T & T & F & F & F & T \\ T & F & T & F & F & T & T & T & F & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$$

$$\begin{array}{|ccccc||ccccc|} \hline p & \lor & (q & \lor & r) & (p & \lor & q) & \lor & r \\ \hline F & F & F & F & F & F & F & F & F & F \\ F & T & F & T & T & F & F & F & T & T \\ F & T & T & T & F & F & T & T & T & F \\ F & T & T & T & T & F & T & T & T & T \\ T & T & F & F & F & T & T & F & T & F \\ T & T & F & T & T & T & T & F & T & T \\ T & T & T & T & F & T & T & T & T & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$$