> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/220-relation-theory/relation-types/reflexive-relation.md). # Reflexive relation ## Reflexive relation In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. Formally, this may be written ∀x ∈ X . (x, x) ∈ R, or as I ⊆ R where I is the identity relation on X. > A binary relation R on set S is **reflexive** if every element of S is related to itself, xRx or R(x,x). R on S is reflexive if `∀a. a ∈ S -> (a,a) ∈ R` A relation $$R$$ on the set $$\mathbb{N^+}$$ is reflexive if it contains **all diagonal (or identity) pairs** e.g. $$R={(1,1),(2,2),(3,3),\dots}$$. ``` ↻◽-------◽↺ | | | | ↻◽-------◽↺ ``` A graph that represents a reflexive relation is the one where each vertex has an identity arrow. ## Irreflexive relation > A binary relation R on set S is **irreflexive** (anti-reflexive) if no element of S is related to itself, ¬R(x,x), x is not R-related to x. Rel R on a set S is irreflexive if `∀a. a ∈ S -> (a,a) ∉ R` A relation $$R$$ on the set $$\mathbb{N^+}$$ is irreflexive if it contains no diagonal (or identity) pairs. ## References