Definition:Inverse Relation
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Definition
Let $\RR \subseteq S \times T$ be a relation.
The inverse relation to (or of) $\RR$ is defined as:
- $\RR^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
That is, $\RR^{-1} \subseteq T \times S$ is the relation which satisfies:
- $\forall s \in S: \forall t \in T: \tuple {t, s} \in \RR^{-1} \iff \tuple {s, t} \in \RR$
Also known as
An inverse relation is also seen as converse relation.
Some sources use the notation $\RR^t$, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra.
Others use $\breve {\RR}$, 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
- Definition:Dual Ordering: the inverse of an ordering.
- Results about inverse relations can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $5.8$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 14$
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.2$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.11$: Relations: Definition $11.4$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.3$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Exercise $4$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.14$