Image of Subset under Composite Relation with Common Codomain and Domain

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Theorem

Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be relations.

Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the composition of $\RR_1$ and $\RR_2$.

Let $A \subseteq S$.


Then:

$\RR_2 \sqbrk {\RR_1 \sqbrk A} = \paren{\RR_2 \circ \RR_1} \sqbrk A$

Proof

We have:

\(\ds \RR_1 \sqbrk A\) \(\subseteq\) \(\ds T\) Image is Subset of Codomain
\(\ds \leadsto \ \ \) \(\ds \RR_1 \sqbrk A\) \(=\) \(\ds \RR_1 \sqbrk A \cap T\) Intersection with Subset is Subset
\(\ds \leadsto \ \ \) \(\ds \RR_2 \sqbrk {\RR_1 \sqbrk A}\) \(=\) \(\ds \RR_2 \sqbrk {\RR_1 \sqbrk A \cap T}\)
\(\ds \) \(=\) \(\ds \paren{\RR_2 \circ \RR_1} \sqbrk A\) Image of Subset under Composite Relation

$\blacksquare$

Also see

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