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An equivalence relation on the natural numbers is co-computably enumerable when its complement as a set of pairs is computably enumerable. Inequivalence can then be positively recognized, even when equivalence cannot be decided. This is sufficient for a semidecidable least-representative transversal.

A congruence on a category $\mathcal{C}$ is an equivalence relation on each hom-set that is compatible with composition on both sides. The quotient category $\mathcal{C}/\sim$ has the same objects and equivalence classes of morphisms, with composition induced from $\mathcal{C}$.

The equivalence class of an element $x$ under an equivalence relation $R$ is the set of all elements related to $x$. Two such sets are equal or disjoint, and together they partition the original set.

Every function $f:X\to Y$ induces an equivalence relation on $X$ by $x\sim y$ exactly when $f(x)=f(y)$. Reflexivity, symmetry, and transitivity follow from the corresponding properties of equality in $Y$; the equivalence classes are the nonempty fibers of $f$.

A groupoid is a category in which every morphism is invertible. A group is a groupoid with one object, while an equivalence relation defines a groupoid with exactly one arrow between any related pair of objects. Every functor sends invertible arrows to invertible arrows.

The intersection of any family of equivalence relations on one set is an equivalence relation. Reflexivity and symmetry hold in every component. A pair of consecutive related points belongs to every component, so each component's transitivity supplies the required final pair. Its equivalence classes refine the component partitions. The empty intersection is the universal relation.