Inverse of Relation Isomorphism is Relation Isomorphism

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Theorem

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

Let $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a bijection.


Then:

$\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$

is a relation isomorphism if and only if:

$\phi^{-1}: \struct {T, \RR_2} \to \struct {S, \RR_1}$

is also a relation isomorphism.


Proof

Follows directly from the definition of relation isomorphism.

$\blacksquare$


Sources