Reflexive relation

In Set theory, a binary Relation can have, inter alia two properties, the reflexivity and the irreflexivity .

a reflexive relation R of the unit X is a relation for which for all has X , has is R - connected to itself. In mathematical Notation, that is written:

$\ forall has \ in X, \ has R a$

a irréflexive relation aliorelative ) --> is a relation for which for all has X , has is never R - connected to itself.

In mathematical Notation, that is written:

$\ forall has \ in X, \ \ lnot (has R a)$ .

Note: Irréflexivité is a condition stronger than the absence of reflexivity. Thus a relation can be reflexive, irréflexive or neither one nor the other. The inequalities " strictly lower à" or " strictly higher à" are irréflexives relations. But, if we define a relation R on the entireties such as has R B if and only if has = - B , then it is neither reflexive, nor irréflexive, because 0 are the only element connected to itself.

Properties containing the reflexive property

Préordre - a reflexive relation which is also transitive. The various types of préordres and relations of equivalence are thus also reflexive.

Examples

Some examples of reflexive relations:

"is equal to " (equality)
" is a Sous-ensemble de" (inclusion of units)
" is larger than or equal à":
" is smaller than or equal à":

Some examples of irréflexives relations:

"à" is not equal;
" is Copremier à"
" que" is strictly larger;:

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