Editing Reflexive relation (section)

=== Related definitions ===

There are several definitions related to the reflexive property. 
The relation <math>R</math> is called:
:; '''{{visible anchor|irreflexive|Irreflexivity|Irreflexive relation}}''', '''{{visible anchor|anti-reflexive|Anti-reflexivity|Anti-reflexive relation}}''' or '''{{visible anchor|aliorelative}}''':<ref>This term is due to [[C S Peirce]]; see {{harvnb|Russell|1920|p=32}}. Russell also introduces two equivalent terms ''to be contained in'' or ''imply diversity''.</ref> if it does not relate any element to itself; that is, if <math>x R x</math> holds for no <math>x \in X.</math> A relation is irreflexive [[if and only if]] its [[Complementary relation|complement]] in <math>X \times X</math> is reflexive. An [[asymmetric relation]] is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric. 
:; '''{{visible anchor|left quasi-reflexive|Left quasi-reflexivity}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x R x.</math><ref name="Britannica">The [https://www.britannica.com/topic/formal-logic/Logical-manipulations-in-LPC#ref534730 Encyclopædia Britannica] calls this property quasi-reflexivity.</ref>
:; '''{{visible anchor|right quasi-reflexive|Right quasi-reflexivity|Right quasi-reflexive relation}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>y R y.</math>
:; '''{{visible anchor|quasi-reflexive|Quasi-reflexivity}}''' : if every element that is part of some relation is related to itself. Explicitly, this means that whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x R x</math> and <math>y R y.</math> Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation <math>R</math> is quasi-reflexive if and only if its [[symmetric closure]] <math>R \cup R^{\operatorname{T}}</math> is left (or right) quasi-reflexive.
:; '''[[Antisymmetric relation|antisymmetric]]''' : if whenever <math>x, y \in X</math> are such that <math>x R y \text{ and } y R x,</math> then necessarily <math>x = y.</math> 
:; '''{{visible anchor|coreflexive|Coreflexivity|Coreflexive relation}}''' : if whenever <math>x, y \in X</math> are such that <math>x R y,</math> then necessarily <math>x = y.</math>{{sfn|ps=|Fonseca de Oliveira|Pereira Cunha Rodrigues|2004|p=337}} A relation <math>R</math> is coreflexive if and only if its symmetric closure is [[Antisymmetric relation|anti-symmetric]].

A reflexive relation on a nonempty set <math>X</math> can neither be irreflexive, nor [[Asymmetric relation|asymmetric]] (<math>R</math> is called {{em|asymmetric}} if <math>x R y</math> implies not <math>y R x</math>), nor [[antitransitive]] (<math>R</math> is {{em|antitransitive}} if <math>x R y \text{ and } y R z</math> implies not <math>x R z</math>).