Symmetric relation

A binary relation R is symmetric if, whenever x is related to y, y is related to x. Formally, xRy implies yRx for every pair in its domain.

Binary relation: A set of ordered pairs that specifies which elements are related to which. Symmetry tests whether reversing every pair preserves membership in this relation.
Equality relation: The relation that holds between two elements exactly when they are equal. Equality is symmetric because x=y always entails y=x.
Reflexive relation: A relation in which every element is related to itself. Reflexivity concerns self-pairs, while symmetry concerns reversing pairs.
Congruence relation: An equivalence relation expressing that mathematical objects have the same specified structure or value. Congruence is symmetric, so either object being congruent to the other gives the reverse relation.
Ordered pair: A pair of objects whose order matters, commonly written (x, y). The symmetry condition compares each pair with its reversal, (y, x).
Divisibility: For integers, a divides b when b is an integer multiple of a. Divisibility is generally asymmetric: 2 divides 4, but 4 does not divide 2.
Antisymmetric relation: A relation where xRy and yRx together imply x=y. Antisymmetry differs from symmetry precisely when distinct elements relate both ways.
Perpendicularity: The relation between lines or directions that meet at a right angle. If one line is perpendicular to another, the reverse statement also holds.
Cartesian product: The set of all ordered pairs formed from elements of two sets. A relation's possible pairs are drawn from a Cartesian product.
Undirected graph: A graph whose edges connect vertices without assigning direction to them. Adjacency in an undirected graph forms a symmetric relation on vertices.

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