Relation Inverse (Converse)

We call the inverse of a relation (in English: converse or inverse relation) the one where we swap every ordered pair of the relation. If (a,b) ∈ R, then (b,a) ∈ R⁻¹.

Formal Definition

The inverse relation thus reverses every connection. For example, if in the 'A is friend of B' relation (A,B) is included, then in the inverse (B,A) will be included.

Examples of Relation Inverses

• If R is the "<" relation on integers, then R⁻¹ is the ">" relation.
• If R is the "parent of" relation among people, then R⁻¹ is the "child of" relation.
• If R is the "teacher of" relation between students and teachers, then R⁻¹ is the "student of" relation.

Properties

• The inverse of the inverse relation is always the original relation: (R⁻¹)⁻¹ = R.
• If a relation is symmetric, it coincides with its own inverse.
• If a relation is transitive, its inverse is also transitive.
• The inverse often gives a different meaning to the connections (e.g., parent of ↔ child of).

Counterexamples and Misconceptions

It is important to distinguish between the inverse of a relation and the complement of a relation. The inverse simply swaps the order of the pairs, while the complement contains all pairs not in the original relation.

Summary

The inverse of a relation reverses the direction of all connections. This is particularly important in mathematics and computer science, where connections often appear as directed edges in graphs. Inverse relations help understand the symmetry and directionality of connections.

Practice Exercise