A binary relation R on sets A and B is a subset of A × B, expressing relationships between elements. Relations have properties: reflexive (relates to itself), symmetric (mutual), transitive (chaining). These properties characterize equivalence and ordering relations.
When you studied the Cartesian product A × B, you saw how to form ordered pairs from two sets. A binary relation is simply a *selection* from those pairs: it picks out which pairs are related. For example, the relation "is a factor of" on the integers selects pairs like (2, 6) and (3, 12) but not (4, 7), because 2 divides 6 but 4 does not divide 7. Formally, R ⊆ A × A (or A × B for a relation between two different sets).
Relations become interesting when we ask about their structural properties. Reflexivity means every element is related to itself — (a, a) ∈ R for all a in A. Equality is reflexive; "is strictly less than" is not. Symmetry means the relation goes both ways: if (a, b) ∈ R then (b, a) ∈ R. "Is a sibling of" is symmetric; "is a parent of" is not. Transitivity means the relation chains: if a R b and b R c, then a R c. "Is an ancestor of" is transitive; "is a friend of" often is not in practice.
These three properties combine in important ways. A relation that is reflexive, symmetric, and transitive is called an equivalence relation — it partitions a set into groups of mutually related elements (equivalence classes). You will study this formally next. A relation that is reflexive, antisymmetric, and transitive is a partial order — like ≤ or "is a subset of" — which captures the idea of ranking or containment without requiring every pair to be comparable.
One subtlety worth watching: the empty relation (no pairs at all) is both symmetric and transitive *vacuously* — there are simply no pairs to violate those conditions. But it is not reflexive on any non-empty set because no element is self-related. This shows that symmetry + transitivity does not automatically give you reflexivity, contrary to a common intuition.