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A binary relation [ilmath]\mathcal{R} [/ilmath] (or just a relation [ilmath]R[/ilmath][Note 1]) between two sets is a subset of the Cartesian product of two sets[1][2], that is:

  • [ilmath]\mathcal{R}\subseteq X\times Y[/ilmath]

We say that [ilmath]\mathcal{R} [/ilmath] is a relation in [ilmath]X[/ilmath][1] if:

  • [ilmath]\mathcal{R}\subseteq X\times X[/ilmath] (note that [ilmath]\mathcal{R} [/ilmath] is sometimes[1] called a graph)
    • For example [ilmath]<[/ilmath] is a relation in the set of [ilmath]\mathbb{Z} [/ilmath] (the integers)

If [ilmath](x,y)\in\mathcal{R} [/ilmath] then we:

  • Say: [ilmath]x[/ilmath] is in relation [ilmath]\mathcal{R} [/ilmath] with [ilmath]y[/ilmath]
  • Write: [ilmath]x\mathcal{R}y[/ilmath] for short.


Here [ilmath]\mathcal{R} [/ilmath] is a relation between [ilmath]X[/ilmath] and [ilmath]Y[/ilmath], that is [ilmath]\mathcal{R}\subseteq X\times Y[/ilmath], and [ilmath]\mathcal{S}\subseteq Y\times Z[/ilmath]

Name Notation Definition
NO IDEA [ilmath]P_X\mathcal{R}[/ilmath][1] [ilmath]P_X\mathcal{R}=\{x\in X\vert\ \exists y:\ x\mathcal{R}y\}[/ilmath] - a function is (among other things) a case where [ilmath]P_Xf=X[/ilmath]
Inverse relation [ilmath]\mathcal{R}^{-1} [/ilmath][1] [ilmath]\mathcal{R}^{-1}:=\{(y,x)\in Y\times X\vert\ x\mathcal{R}y\}[/ilmath]
Composing relations [ilmath]\mathcal{R}\circ\mathcal{S} [/ilmath][1] [ilmath]\mathcal{R}\circ\mathcal{S}:=\{(x,z)\in X\times Z\vert\ \exists y\in Y[x\mathcal{R}y\wedge y\mathcal{S}z]\}[/ilmath]

Simple examples of relations

  1. The empty relation[1], [ilmath]\emptyset\subset X\times X[/ilmath] is of course a relation
  2. The total relation[1], [ilmath]\mathcal{R}=X\times X[/ilmath] that relates everything to everything
  3. The identity relation[1], [ilmath]\text{id}_X:=\text{id}:=\{(x,y)\in X\times X\vert x=y\}=\{(x,x)\in X\times X\vert x\in X\}[/ilmath]
    • This is also known as[1] the diagonal of the square [ilmath]X\times X[/ilmath]

Types of relation

Here [ilmath]\mathcal{R}\subseteq X\times X[/ilmath]

Name Set relation Statement Notes
Reflexive[1] [ilmath]\text{id}_X\subseteq\mathcal{R} [/ilmath] [ilmath]\forall x\in X[x\mathcal{R}x][/ilmath] Every element is related to itself (example, equality)
Symmetric[1] [ilmath]\mathcal{R}\subseteq\mathcal{R}^{-1} [/ilmath] [ilmath]\forall x\in X\forall y\in X[x\mathcal{R}y\implies y\mathcal{R}x][/ilmath] (example, equality)
Transitive[1] [ilmath]\mathcal{R}\circ\mathcal{R}\subseteq\mathcal{R} [/ilmath] [ilmath]\forall x,y,z\in X[(x\mathcal{R}y\wedge y\mathcal{R}z)\implies x\mathcal{R}z][/ilmath] (example, equality, [ilmath]<[/ilmath])
(AKA Identitive[1])
[ilmath]\mathcal{R}\cap\mathcal{R}^{-1}\subseteq\text{id}_X[/ilmath] [ilmath]\forall x\in X\forall y\in X[(x\mathcal{R}y\wedge y\mathcal{R}x)\implies x=y][/ilmath]

TODO: What about a relation like 1r2 1r1 2r1 and 2r2

Connected[1] [ilmath]\mathcal{R}\cup\mathcal{R}^{-1}=X\times X[/ilmath]

TODO: Work out what this means

Asymmetric[1] [ilmath]\mathcal{R}\subseteq\complement(\mathcal{R}^{-1})[/ilmath] [ilmath]\forall x\in X\forall y\in X[x\mathcal{R}y\implies (y,x)\notin\mathcal{R}][/ilmath] Like [ilmath]<[/ilmath] (see: Contrapositive)
Right-unique[1] [ilmath]\mathcal{R}^{-1}\circ\mathcal{R}\subseteq\text{id}_X[/ilmath] [ilmath]\forall x,y,z\in X[(x\mathcal{R}y\wedge x\mathcal{R}z)\implies y=z][/ilmath] This is the definition of a function
Left-unique[1] [ilmath]\mathcal{R}\circ\mathcal{R}^{-1}\subseteq\text{id}_X[/ilmath] [ilmath]\forall x,y,z\in X[(x\mathcal{R}y\wedge z\mathcal{R}y)\implies x=z][/ilmath]
Mutually unique[1] Both right and left unique

TODO: Investigate

Examples of binary relations


  1. A binary relation should be assumed if just relation is specified


  1. 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 Analysis - Part 1: Elements - Krzysztof Maurin
  2. Types and Programming Languages - Benjamin C. Peirce
  3. Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg