Composition of Relations is Associative

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Theorem

The composition of relations is an associative binary operation:

$\paren {\RR_3 \circ \RR_2} \circ \RR_1 = \RR_3 \circ \paren {\RR_2 \circ \RR_1}$


Proof

First, note that from the definition of composition of relations, the following must be the case before the above expression is even to be defined:

$\Dom {\RR_2} = \Cdm {\RR_1}$
$\Dom {\RR_3} = \Cdm {\RR_2}$


The two composite relations can be seen to have the same domain, thus:

\(\displaystyle \Dom {\paren {\RR_3 \circ \RR_2} \circ \RR_1}\) \(=\) \(\displaystyle \Dom {\RR_1}\) Domain of Composite Relation


\(\displaystyle \Dom {\RR_3 \circ \paren {\RR_2 \circ \RR_1} }\) \(=\) \(\displaystyle \Dom {\RR_2 \circ \RR_1}\) Domain of Composite Relation
\(\displaystyle \) \(=\) \(\displaystyle \Dom {\RR_1}\) Domain of Composite Relation


... and also the same codomain, thus:

\(\displaystyle \Cdm {\paren {\RR_3 \circ \RR_2} \circ \RR_1}\) \(=\) \(\displaystyle \Cdm {\RR_3 \circ \RR_2}\) Codomain of Composite Relation
\(\displaystyle \) \(=\) \(\displaystyle \Cdm {\RR_3}\) Codomain of Composite Relation


\(\displaystyle \Cdm {\RR_3 \circ \paren {\RR_2 \circ \RR_1} }\) \(=\) \(\displaystyle \Cdm {\RR_3}\) Codomain of Composite Relation


So they are equal if and only if they have the same value at each point in their common domain, which this shows:

\(\displaystyle \forall x \in \Dom {\RR_1}: \ \ \) \(\displaystyle \map {\paren {\paren {\RR_3 \circ \RR_2} \circ \RR_1} } x\) \(=\) \(\displaystyle \map {\paren {\RR_3 \circ \RR_2} } {\map {\RR_1} x}\) Definition of Composition of Relations
\(\displaystyle \) \(=\) \(\displaystyle \map {\RR_3} {\map {\RR_2} {\map {\RR_1} x} }\) Definition of Composition of Relations
\(\displaystyle \) \(=\) \(\displaystyle \map {\RR_3} {\map {\paren {\RR_2 \circ \RR_1} } x}\) Definition of Composition of Relations
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {\RR_3 \circ \paren {\RR_2 \circ \RR_1} } } x\) Definition of Composition of Relations

$\blacksquare$


Sources