xRx

xRx is a concise notation used in mathematics and related disciplines to indicate that a pair consisting of an element x with itself is in a binary relation R defined on a set X. In this sense, xRx is equivalent to saying that (x, x) ∈ R. The statement is commonly used when discussing properties of R, such as reflexivity, symmetry, and transitivity.

In the context of a relation R on a set X, the expression xRx is most directly associated with reflexivity. A relation is reflexive if every element relates to itself, i.e., for all x in X, xRx holds. Classic examples include the equality relation on any set, where x = x is always true; in that case the diagonal Δ = {(x, x) : x ∈ X} is a subset of R (indeed, Δ ⊆ R when R is reflexive).

In graph-theoretic terms, a relation R on a set of vertices V can be viewed as a directed graph with an edge from x to y whenever (x, y) ∈ R. The condition xRx corresponds to a loop at vertex x. The presence or absence of self-edges is a standard way to describe a relation’s reflexive property and its other structural characteristics.

xRx also appears in discussions of diagonal concepts in linear algebra and topology, where the diagonal elements or diagonal relations play a role in defining projections, kernels, or product constructions. When R is defined via a property P such that xRx means x has property P, the phrase serves as a compact shorthand in proofs and definitions.