Editing Binary relation (section)

=== Contact ===
Suppose <math>B</math> is the [[power set]] of <math>A</math>, the set of all [[subset]]s of <math>A</math>. Then a relation <math>g</math> is a '''contact relation''' if it satisfies three properties: 
# <math>\text{for all } x \in A, Y = \{ x \} \text{ implies } xgY.</math>
# <math>Y \subseteq Z \text{ and } xgY \text{ implies } xgZ.</math>
# <math>\text{for all } y \in Y, ygZ \text{ and } xgY \text{ implies } xgZ.</math>
The [[set membership]] relation, <math>\epsilon = </math> "is an element of", satisfies these properties so <math>\epsilon</math> is a contact relation. The notion of a general contact relation was introduced by [[Georg Aumann]] in 1970.<ref>{{cite journal | url=https://www.zobodat.at/publikation_volumes.php?id=56359 | author=Georg Aumann | title=Kontakt-Relationen | journal=Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München | volume=1970 | number=II | pages=67&ndash;77 | year=1971 }}</ref><ref>Anne K. Steiner (1970) [https://mathscinet.ams.org/mathscinet-getitem?mr=0309040 Review:''Kontakt-Relationen''] from [[Mathematical Reviews]]</ref>

In terms of the calculus of relations, sufficient conditions for a contact relation include
<math display="block">C^\textsf{T} \bar{C} \subseteq \ni \bar{C} \equiv C \overline{\ni \bar{C}} \subseteq C,</math> 
where <math>\ni</math> is the converse of set membership (<math>\in</math>).<ref name=GS11/>{{rp|280}}