Relations, Orders and Equivalence Classes

A relation on a set is simply a rule for which pairs of elements are considered "related" — nothing more than a chosen collection of pairs. Two families of relations show up constantly in quantitative work: equivalence relations, which capture "these two things count as the same for our purposes," and order relations, which capture "this thing ranks above that one."

An equivalence relation must satisfy three properties: reflexivity (every element relates to itself), symmetry (if $a$ relates to $b$, then $b$ relates to $a$), and transitivity (if $a$ relates to $b$ and $b$ relates to $c$, then $a$ relates to $c$). "Same sign" on the real numbers is an equivalence relation; it partitions all numbers into disjoint groups called equivalence classes — here, the positives, the negatives, and zero — where every element in a class is treated as interchangeable with every other element of that class for the purpose at hand.

An order relation instead requires reflexivity, transitivity, and antisymmetry (if $a$ relates to $b$ and $b$ relates to $a$, then $a$ and $b$ must actually be equal), and it need not rank every pair — a "partial order" allows some elements to simply be incomparable. Ordinary $\leq$ on the real numbers is a "total order" because any two numbers are comparable; but portfolio dominance orderings (one portfolio dominates another only if it beats it on every risk measure simultaneously) are typically only partial orders, since two portfolios can each beat the other on a different measure and so are genuinely incomparable rather than tied.