Symmetric relation
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.
In mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, this is:
Note: symmetry is not the exact opposite of antisymmetry
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
(aRb and bRa implies b = a).
