Reflexive relation

Author: Leandro Alegsa Creation date: November 13, 2022

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A relation is said to be reflexive when for all members of the relations R, x=x.

For example, for the set A, which only includes the ordered pair (1,1). The relation is reflexive as 1=1.

Formal definition

If a set and $R\subseteq M\times M$ a two-digit relation on , then one defines (using infix notation):

is reflexive : $\Longleftrightarrow \forall x\in M:xRx$

is irreflexive : $\Longleftrightarrow \forall x\in M:\neg \ xRx$

The less than or equal relation ≤ $\leq$ on the real numbers is reflexive, since always $x\leq x$ holds. Moreover, it is a total order. The same holds for the relation ≥ $\geq$ .

The ordinary equality on the real numbers is reflexive, since always holds. Moreover, it is an equivalence relation.

The subset relation $\subseteq$ between sets is reflexive, since always $A\subseteq A$ holds. Moreover, it is a half-order.

The Kleiner relation on the real numbers is irreflexive, since never holds. Moreover, it is a strict total order. The same is true for the relation .

The inequality $\ne$ on the real numbers is irreflexive, since never $x\neq x$ holds.

The real subset relation $\subset$ between sets is irreflexive, since never $A\subset A$ holds. Moreover, it is a strict half-order.

The following relation on the set of real numbers is neither reflexive nor irreflexive:

$xRy:\Longleftrightarrow y=x^{2}$

Reason: For holds xRx , for holds ¬ $\neg xRx$ .