Editing Binary relation (section)

== Examples ==
{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"
|+ 2nd example relation
! {{diagonal split header|<math>B</math>|<math>A</math>}}
! scope="col" | ball
! scope="col" | car
! scope="col" | doll
! scope="col" | cup
|-
! scope="row" | John 
| '''+''' || − || − || −
|-
! scope="row" | Mary 
| − || − || '''+''' || −
|-
! scope="row" | Venus 
| − || '''+''' || − || −
|}
{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"
|+ 1st example relation
! {{diagonal split header|<math>B</math>|<math>A</math>}}
! scope="col" | ball
! scope="col" | car
! scope="col" | doll
! scope="col" | cup
|-
! scope="row" | John 
| '''+''' || − || − || −
|-
! scope="row" | Mary 
| − || − || '''+''' || −
|-
! scope="row" | Ian 
| − || − || − || −
|-
! scope="row" | Venus 
| − || '''+''' || − || −
|}
{{olist
|1= The following example shows that the choice of codomain is important. Suppose there are four objects <math>A = \{ \text{ball, car, doll, cup} \}</math> and four people <math>B = \{ \text{John, Mary, Ian, Venus} \}.</math> A possible relation on <math>A</math> and <math>B</math> is the relation "is owned by", given by <math>R = \{ (\text{ball, John}), (\text{doll, Mary}), (\text{car, Venus}) \}.</math> That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set, <math>R</math> does not involve Ian, and therefore <math>R</math> could have been viewed as a subset of <math>A \times \{ \text{John, Mary, Venus} \},</math> i.e. a relation over <math>A</math> and <math>\{ \text{John, Mary, Venus} \};</math> see the 2nd example. But in that second example, <math>R</math> contains no information about the ownership by Ian.

While the 2nd example relation is surjective (see [[#Types of binary relations|below]]), the 1st is not.

[[File:Oceans and continents coarse.png|thumb|250px|right|Oceans and continents (islands omitted)]]
{{pipe escape|
{| class{{=}}"wikitable" style{{=}}"float: right; margin-left:1em; text-align:center;"
|+Ocean borders continent
!
! scope{{=}}"col" | NA
! scope{{=}}"col"      | SA
! scope{{=}}"col"            | AF
! scope{{=}}"col"                 | EU
! scope{{=}}"col"                       | AS
! scope{{=}}"col"                             | AU
! scope{{=}}"col"                                  | AA
|-
! scope{{=}}"row" | Indian
|0||0||1||0||1||1||1
|-  
! scope{{=}}"row" | Arctic
|1||0||0||1||1||0||0 
|-
! scope{{=}}"row" | Atlantic
|1||1||1||1||0||0||1
|-
! scope{{=}}"row" | Pacific
|1||1||0||0||1||1||1
|}
}}

|2= Let <math>A = \{\text{Indian}, \text{Arctic}, \text{Atlantic}, \text{Pacific}\}</math>, the [[ocean]]s of the globe, and <math>B = \{\text{NA}, \text{SA}, \text{AF}, \text{EU}, \text{AS}, \text{AU}, \text{AA}\}</math>, the [[continent]]s. Let <math>aRb</math> represent that ocean <math>a</math> borders continent <math>b</math>. Then the [[logical matrix]] for this relation is:
:<math>R = \begin{pmatrix} 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 & 1 \end{pmatrix} .</math>
The connectivity of the planet Earth can be viewed through <math>R R^\mathsf{T}</math> and <math>R^\mathsf{T} R</math>, the former being a <math>4 \times 4</math> relation on <math>A</math>, which is the universal relation (<math>A \times A</math> or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, <math>R^\mathsf{T} R</math> is a relation on <math>B \times B</math> which ''fails'' to be universal because at least two oceans must be traversed to voyage from [[Europe]] to [[Australia]].

|3= Visualization of relations leans on [[graph theory]]: For relations on a set (homogeneous relations), a [[directed graph]] illustrates a relation and a [[graph (discrete mathematics)|graph]] a [[symmetric relation]]. For heterogeneous relations a [[hypergraph]] has edges possibly with more than two nodes, and can be illustrated by a [[bipartite graph]].

Just as the [[clique (graph theory)|clique]] is integral to relations on a set, so [[biclique]]s are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.
[[File:Add_velocity_ark_POV.svg|right|thumb|200px|The various <math>t</math> axes represent time for observers in motion, the corresponding <math>x</math> axes are their lines of simultaneity]]

|4= [[Hyperbolic orthogonality]]: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of {{em|simultaneous events}} is simple in [[absolute time and space]] since each time <math>t</math> determines a simultaneous [[hyperplane]] in that cosmology. [[Hermann Minkowski]] changed that when he articulated the notion of {{em|relative simultaneity}}, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a [[composition algebra]] is given by
:<math>\langle x, z\rangle = x \bar{z} + \bar{x}z\;</math> where the overbar denotes conjugation.
As a relation between some temporal events and some spatial events, [[hyperbolic orthogonality]] (as found in [[split-complex number]]s) is a heterogeneous relation.<ref>{{wikibooks inline|Calculus/Hyperbolic angle#Split-complex theory|Relative simultaneity}}</ref>

|5= A [[geometric configuration]] can be considered a relation between its points and its lines. The relation is expressed as [[incidence relation|incidence]]. Finite and infinite projective and affine planes are included. [[Jakob Steiner]] pioneered the cataloguing of configurations with the [[Steiner system]]s <math>\operatorname S(t, k, n)</math> which have an n-element set <math>\operatorname S</math> and a set of k-element subsets called '''blocks''', such that a subset with <math>t</math> elements lies in just one block. These [[incidence structure]]s have been generalized with [[block design]]s. The [[incidence matrix]] used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
: An incidence structure is a triple <math>\mathbf D = (V, \mathbf B, I)</math> where <math>V</math> and <math>\mathbf B</math> are any two disjoint sets and <math>I</math> is a binary relation between <math>V</math> and <math>\mathbf B</math>, i.e. <math>I \subseteq V \times \mathbf B.</math> The elements of <math>V</math> will be called {{em|points}}, those of <math>\mathbf B</math> {{em|blocks}}, and those of <math>I</math> {{em|flags}}.<ref>{{cite book|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|authorlink2=Dieter Jungnickel|first3=Hanfried|last3=Lenz|authorlink3=Hanfried Lenz|title=Design Theory|publisher=[[Cambridge University Press]]|page=15|year=1986}}. 2nd ed. (1999) {{ISBN|978-0-521-44432-3}}</ref>
}}