Rule of Association

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Theorem

Conjunction

Conjunction is associative:

Formulation 1

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

Formulation 2

$\vdash \paren {p \land \paren {q \land r} } \iff \paren {\paren {p \land q} \land r}$


Disjunction

Disjunction is associative:

Formulation 1

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

Formulation 2

$\vdash \paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$


Also known as

The rule of association is also known as the associative law.


Note that this term is also used throughout mathematics in the context of addition and multiplication of numbers:

Associative Law of Addition
Associative Law of Multiplication

so it is wise to be aware of the context.


Also see


Sources