Definition:Inverse Relation

From ProofWiki
Jump to navigation Jump to search

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:

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


Class Theoretical Definition

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $A$ and $B$ be subclasses of $V$.

Let $\RR \subseteq A \times B$ be a relation on $A \times B$.


The inverse relation to (or of) $\RR$ is defined as the class of all ordered pairs $\tuple {b, a}$ such that $\tuple {a, b} \in \RR$:

$\RR^{-1} := \set {\tuple {b, a}: \tuple {a, b} \in \RR}$


Examples

Divisor

Let $\Z_{>0}$ denote the set of (strictly) positive integers.

Let $\RR$ denote the relation on $\Z_{>0}$ defined as:

$\forall a, b \in \Z_{>0}: \tuple {a, b} \in \RR \iff a \divides b$

where $a \divides b$ denotes that $a$ is a divisor of $b$.


The inverse relation of $\RR$ is defined as:

$\forall a, b \in \Z_{>0}: \tuple {a, b} \in \RR^{-1} \iff b \divides a$

That is, if and only if $a$ is a multiple of $b$.


Also known as

An inverse relation is also seen as converse relation.

In the context of orderings, the term dual ordering can also be seen.

However, do not confuse this with a dual relation.


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

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

Still others use $\RR^\gets$, for example Seth Warner: Modern Algebra.


Also see

  • Results about inverse relations can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Inverse_Relation&oldid=715905"