Composition of Relations is Associative
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
 | This has to be rewritten. In particular: Needs to be rewritten in light of the actual definition of the composition of relations and domain of relations. These equalities may not hold for all relations You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by doing so. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Rewrite}} from the code. |
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:
| \(\ds \Dom {\paren {\RR_3 \circ \RR_2} \circ \RR_1}\) | \(=\) | \(\ds \Dom {\RR_1}\) | Domain of Composite Relation |
| \(\ds \Dom {\RR_3 \circ \paren {\RR_2 \circ \RR_1} }\) | \(=\) | \(\ds \Dom {\RR_2 \circ \RR_1}\) | Domain of Composite Relation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \Dom {\RR_1}\) | Domain of Composite Relation |
and also the same codomain, thus:
| \(\ds \Cdm {\paren {\RR_3 \circ \RR_2} \circ \RR_1}\) | \(=\) | \(\ds \Cdm {\RR_3 \circ \RR_2}\) | Codomain of Composite Relation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \Cdm {\RR_3}\) | Codomain of Composite Relation |
| \(\ds \Cdm {\RR_3 \circ \paren {\RR_2 \circ \RR_1} }\) | \(=\) | \(\ds \Cdm {\RR_3}\) | Codomain of Composite Relation |
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So they are equal if and only if they have the same value at each point in their common domain, which this shows:
| This has to be rewritten. In particular: These terms may not be defined for all relations. So a different approach is required You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by doing so. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Rewrite}} from the code. |
| \(\ds \forall x \in \Dom {\RR_1}: \, \) | \(\ds \map {\paren {\paren {\RR_3 \circ \RR_2} \circ \RR_1} } x\) | \(=\) | \(\ds \map {\paren {\RR_3 \circ \RR_2} } {\map {\RR_1} x}\) | Definition of Composition of Relations | ||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\RR_3} {\map {\RR_2} {\map {\RR_1} x} }\) | Definition of Composition of Relations | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\RR_3} {\map {\paren {\RR_2 \circ \RR_1} } x}\) | Definition of Composition of Relations | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\paren {\RR_3 \circ \paren {\RR_2 \circ \RR_1} } } x\) | Definition of Composition of Relations |
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $5 \ \text{(b)}$
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions: Exercise $5.8 \ \text{(b)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text I$: Sets and Functions: Problem $\text{AA}$: Relations