Inverse of Subset of Relation is Subset of Inverse

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Theorem

Let $S$ and $T$ be sets

Let $\RR_1 = S \times T$ be a relation on $S \times T$.

Let $\RR_2 \subseteq \RR_1$.


Then:

$\RR_2^{-1} \subseteq \RR_1^{-1}$

where $\RR_1^{-1}$ denotes the inverse of $\RR_1$.


Proof

\(\ds \tuple {t, s}\) \(\in\) \(\ds \RR_2^{-1}\)
\(\ds \leadsto \ \ \) \(\ds \tuple {s, t}\) \(\in\) \(\ds \RR_2\) Definition of Inverse Relation
\(\ds \leadsto \ \ \) \(\ds \tuple {s, t}\) \(\in\) \(\ds \RR_1\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds \tuple {t, s}\) \(\in\) \(\ds \RR_1^{-1}\) Definition of Inverse Relation

The result follows by definition of subset.

$\blacksquare$