Relations and Orders glossary

Clear, one-line definitions of the Relations and Orders terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

antisymmetric

A relation R where aRb and bRa together imply a = b; partial orders are antisymmetric.

composition of relations

For R ⊆ A × B and S ⊆ B × C: S ∘ R = {(a, c) : ∃b, (a, b) ∈ R and (b, c) ∈ S}.

domain of a relation

For R ⊆ A × B: dom(R) = {a ∈ A : ∃b, (a, b) ∈ R}; the set of first coordinates.

equivalence class

For an equivalence relation R: [a] = {x : aRx}; the set of all elements R-related to a.

equivalence relation

A reflexive, symmetric, and transitive relation; it partitions its underlying set into equivalence classes.

See: Equivalence Relations

hasse diagram

A graphical representation of a finite poset showing only the cover relations (no transitive edges drawn).

inequality

A statement comparing two expressions with <, ≤, >, or ≥ instead of equality.

interval

A connected subset of the real line, such as [a, b], (a, b), or [a, ∞).

See: Stacking, RMS, and interval velocities, Prediction intervals vs confidence intervals

inverse relation

R⁻¹ = {(b, a) : (a, b) ∈ R}; swap each ordered pair.

partial order

A reflexive, antisymmetric, and transitive relation; not every pair need be comparable.

range of a relation

For R ⊆ A × B: ran(R) = {b ∈ B : ∃a, (a, b) ∈ R}; the set of second coordinates.

reflexive

A relation R is reflexive when aRa holds for every a in the underlying set.

relation

A subset of A × B (or A × A); specifies which ordered pairs are 'related.'

See: Relations, Ordering Relations

symmetric

A relation R where aRb implies bRa.

total order

A partial order in which every pair of elements is comparable: aRb or bRa for all a, b.

transitive

A relation R where aRb and bRc together imply aRc.

transitive closure

The smallest transitive relation containing R; add (a, c) whenever a chain a → b → ... → c exists in R.