Relations

Precise definition

A binary relation $R$ from $A$ to $B$ is a subset of $A\times B$; write $aRb$ when $(a,b)\in R$. A relation on $A$ is reflexive if $aRa$ for all $a$, symmetric if $aRb\Rightarrow bRa$, antisymmetric if $aRb$ and $bRa$ imply $a=b$, and transitive if $aRb$ and $bRc$ imply $aRc$.

Notation and mathematical language

An equivalence relation is reflexive, symmetric, and transitive; it partitions $A$ into equivalence classes $[a]=\{x:xRa\}$. A partial order is reflexive, antisymmetric, and transitive. Symmetric and antisymmetric are not opposites: equality satisfies both.

Conceptual picture

A relation is a selected set of arrows between elements. Matrices and directed graphs display finite relations; property tests become patterns involving loops, reversed arrows, and completed two-step paths. Equivalence classes group indistinguishable elements under the chosen criterion.

Fully worked example

Interpretation and application

Relations model prerequisites, database links, rankings, congruence, and state reachability. A partial order may leave elements incomparable; interpreting it as a complete ranking adds information the relation does not contain.

Common mistakes