Lecture
A reflexive relation in mathematics — is a binary relation 
on a set 
, under which every element of this set is in the relation with itself .
Formally, the relation is reflexive if 
.
The word «reflexive» originally comes from the Medieval Latin reflexivus («turning back» [cf. reflex ], or «directed at itself») (c. 1250 A.D.) from the Classical Latin reflexus- («turning back», «reflection») + -īvus (suffix).
The reflexivity property of a relation, when given by a matrix, is characterized by the fact that all diagonal elements of the matrix equal 1; when the relation is given by a graph, each element x has a loop — an arc (x, x).
A binary relation on a set is reflexive if and only if its subset is the identity relation 
on the set 
), that is,
.
If 
is meaningless, then the relation is called anti-reflexive (or irreflexive)[1].
If an anti-reflexive relation is given by a matrix, then all diagonal elements are zero. When such a relation is given by a graph, each vertex has no loop — there are no arcs of the form (x, x).
Formally, the anti-reflexivity of a relation is defined as: 
.
If the reflexivity condition is not satisfied for all elements of the set , the relation is said to be non-reflexive.
Reflexive relations:
);
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).
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);Authors in philosophical logic often use different terminology. Reflexive relations in the mathematical sense are called totally reflexive in philosophical logic, and quasi-reflexive relations are called reflexive .
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