Difference between revisions of "Reflexivity"
From Encyclopedia of Mathematics
(Category:Logic and foundations) |
(→References: isbn link) |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{TEX|done}} | {{TEX|done}} | ||
| − | A property of binary relations. A [[binary relation]] $R$ on a set $A$ is called reflexive if $aRa$ for all $a\in A$. Examples of reflexive relations are equality (cf [[Equality axioms]]), [[equivalence relation]]s, [[Order (on a set)|order]]. | + | A property of binary relations. A [[binary relation]] $R$ on a set $A$ is called reflexive if $aRa$ for all $a\in A$. Regarding $R$ as a subset of $A \times A$, $R$ is reflexive if it contains the diagonal or identity relation $\Delta = \{(a,a) : a \in A \}$. Examples of reflexive relations are equality (cf [[Equality axioms]]), [[equivalence relation]]s, [[Order (on a set)|order]]. |
====References==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) ISBN 0080960413</TD></TR> | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) {{ISBN|0080960413}}</TD></TR> |
| − | <TR><TD valign="top">[a2]</TD> <TD valign="top"> P. R. Halmos, ''Naive Set Theory'', Springer (1960) ISBN 0-387-90092-6</TD></TR> | + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> P. R. Halmos, ''Naive Set Theory'', Springer (1960, repr. 1974) {{ISBN|0-387-90092-6}} {{ZBL|0287.04001}}</TD></TR> |
</table> | </table> | ||
[[Category:Logic and foundations]] | [[Category:Logic and foundations]] | ||
Latest revision as of 19:34, 17 November 2023
A property of binary relations. A binary relation $R$ on a set $A$ is called reflexive if $aRa$ for all $a\in A$. Regarding $R$ as a subset of $A \times A$, $R$ is reflexive if it contains the diagonal or identity relation $\Delta = \{(a,a) : a \in A \}$. Examples of reflexive relations are equality (cf Equality axioms), equivalence relations, order.
References
| [a1] | R. Fraïssé, Theory of Relations, Studies in Logic and the Foundations of Mathematics, Elsevier (2011) ISBN 0080960413 |
| [a2] | P. R. Halmos, Naive Set Theory, Springer (1960, repr. 1974) ISBN 0-387-90092-6 Zbl 0287.04001 |
How to Cite This Entry:
Reflexivity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflexivity&oldid=33961
Reflexivity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflexivity&oldid=33961
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article