Definition:Inverse Relation

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Let $\mathcal R \subseteq S \times T$ be a relation.

The inverse relation to (or of) $\mathcal R$ is defined as:

$\mathcal R^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in \mathcal R}$

That is, $\mathcal R^{-1} \subseteq T \times S$ is the relation which satisfies:

$\forall s \in S: \forall t \in T: \tuple {t, s} \in \mathcal R^{-1} \iff \tuple {s, t} \in \mathcal R$

Also known as

An inverse relation is also seen as converse relation.

Some sources use the notation $\mathcal R^t$, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra.

Others use $\breve {\mathcal R}$, for example 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) and 1940: Garrett Birkhoff: Lattice Theory.

Also see

  • Results about inverse relations can be found here.