# Reflexive relation

## Binary relation that relates every element to itself / From Wikipedia, the free encyclopedia

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In mathematics, a binary relation $R$ on a set $X$ is **reflexive** if it relates every element of $X$ to itself.^{[1]}^{[2]}

Transitive binary relations | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ |

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the **reflexive property** or is said to possess **reflexivity**. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.