Arb

In mathematics, the notation aRb is used to denote that elements a and b are related by a binary relation R. When R is a relation on a set X, a and b are elements of X and the statement aRb means that the ordered pair (a,b) belongs to R, i.e., (a,b) ∈ R.

Relations, and thus the expression aRb, can have various properties. A relation is reflexive if every element relates to itself (aRa for all a). It is symmetric if aRb implies bRa. It is antisymmetric if aRb and bRa together imply a = b. It is transitive if aRb and bRc imply aRc. A relation that is reflexive, symmetric, and transitive is an equivalence relation, while a reflexive and transitive relation is a partial order. These properties help classify and analyze structures in mathematics.

Common examples illustrate the idea. The equality relation aRb with R = { (a,a) | a in X } yields a = b. The usual order relation a ≤ b on real numbers yields aRb when a ≤ b. The divisibility relation a | b on integers yields aRb when a divides b. In graph theory, aRb corresponds to a directed edge from a to b in a directed graph representing the relation.

Notes and variants: In logic, R(a,b) is a predicate with two arguments. If R is a function rather than a relation, one typically writes b = f(a) rather than aRb. The concept of aRb underlies many areas, including set theory, database relations, and the study of ordered or partitioned structures.