Inverse Relational Structures of Isomorphic Structures are Isomorphic

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Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures.

Let ${\RR_1}^{-1}$ and ${\RR_2}^{-1}$ be the inverses of $\RR_1$ and $\RR_2$ respectively.

Let $f: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a relation isomorphism.

Then $f: \struct {S, {\RR_1}^{-1} } \to \struct {T, {\RR_2}^{-1} } $ is also a relation isomorphism.


\(\ds \forall x, y \in S: \, \) \(\ds x\) \({\RR_1}^{-1}\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds y\) \(\RR_1\) \(\ds x\) Definition of Inverse Relation
\(\ds \leadstoandfrom \ \ \) \(\ds \map f y\) \(\RR_2\) \(\ds \map f x\) Definition of Relation Isomorphism
\(\ds \leadstoandfrom \ \ \) \(\ds \map f x\) \({\RR_2}^{-1}\) \(\ds \map f y\) Definition of Inverse Relation