Relations.html (2704B)
1 <?xml version="1.0" encoding="UTF-8"?> 2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> 3 <html><head><link rel="stylesheet" type="text/css" href="sitewide.css"><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/><meta name="exporter-version" content="Evernote Mac 7.0.3 (456341)"/><meta name="keywords" content="sets"/><meta name="altitude" content="-1.882289528846741"/><meta name="author" content="Alex Balgavy"/><meta name="created" content="2018-02-15 14:42:35 +0000"/><meta name="latitude" content="52.33467867860909"/><meta name="longitude" content="4.866408472356967"/><meta name="source" content="desktop.mac"/><meta name="updated" content="2018-02-15 16:13:50 +0000"/><title>Relations</title></head><body><div><b>Cartesian product of sets</b></div><div><br/></div><div>A × B := {<a,b> : a ∈ A ∩ b ∈ B}</div><div>A × A := A<sup>2</sup></div><div><sup><br/></sup></div><div>#(A×B) = #A ⋅ #B</div><div><br/></div><div><b>Binary relation:</b> relation of type A × B or A × A</div><div><b>Relation in set A:</b> relation of type A × A</div><div><br/></div><div><b>Infix notation:</b></div><div>x R y — <x, y> ∈ R</div><div><br/></div><div><br/></div><div><b>Visualisation</b></div><div>directed graphs & matrix:</div><div><br/></div><div><br/></div><div><img src="Relations.resources/screenshot_2.png" height="301" width="619"/></div><div><br/></div><div>Venn diagrams & matrix:</div><div><br/></div><div><img src="Relations.resources/screenshot_1.png" height="301" width="627"/></div><div><br/></div><div><b>Inverse of binary relation</b></div><div>Inverse of R: R<sup>-1</sup> := {<x,y> : <y,x> ∈ R}</div><div>R ⊆ A × B => R<sup>-1</sup> ⊆ B × A</div><div><br/></div><div>For Venn diagrams, you reverse the arrows.</div><div><br/></div><div><b>Composite relations</b></div><div>R ∘ S := {<x,z> : x S y ∩ y R z for some y}</div><div><br/></div><div><img src="Relations.resources/screenshot_3.png" height="209" width="644"/></div><div><br/></div><div>Composition is associative.</div><div>Inverse: (R ∘ S)<sup style="font-size: 11.666666030883789px;">-1</sup><span style="font-size: 11.666666030883789px;"> </span><font style="font-size: 14px;">= S<sup>-1</sup> ∘ R<sup>-1</sup></font></div><div><font style="font-size: 14px;"><sup><br/></sup></font></div><div><font style="font-size: 14px;"><b>Properties of relations</b></font></div><div><img src="Relations.resources/screenshot.png" height="372" width="635"/></div><div><img src="Relations.resources/screenshot_4.png" height="372" width="635"/></div><div><br/></div></body></html>