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A binary relation R on a set S is a subset R ⊆ S × S. Relations exhibit properties including reflexivity (aRa for all a), symmetry (aRb ⟹ bRa), and transitivity (aRb ∧ bRc ⟹ aRc). Different combinations of these properties define important relation classes: equivalence relations, orderings, and more.
From your work with ordered pairs, you know that S × S is the set of all pairs (a, b) where a and b come from S. A binary relation R on S is simply a subset of this Cartesian product: a collection of pairs. When (a, b) ∈ R, we write aRb and say "a is related to b." This set-theoretic definition is completely general — any collection of pairs is a valid relation. The interesting structure comes from which properties that collection happens to satisfy.
Three properties appear most often. Reflexivity says every element is related to itself: for all a ∈ S, aRa. The equality relation on any set is reflexive; so is "less than or equal to" on the integers. Symmetry says the relation runs in both directions: if aRb then bRa. "Is a sibling of" is symmetric; "is a parent of" is not. Transitivity says the relation chains: if aRb and bRc then aRc. "Is an ancestor of" is transitive; "is a neighbor of" (in a graph) typically is not. A fourth property appears in orderings: antisymmetry says that if aRb and bRa, then a = b — the relation can go both ways only when the two elements are the same.
These properties combine to define the most important relation types in mathematics. Equivalence relations satisfy reflexivity, symmetry, and transitivity — they partition S into equivalence classes where all related elements are grouped together. "Has the same remainder when divided by 3" is an equivalence relation on the integers, grouping them into classes {0, 3, 6, ...}, {1, 4, 7, ...}, {2, 5, 8, ...}. Partial orders satisfy reflexivity, antisymmetry, and transitivity — they give a notion of "at most as large as" without requiring all elements to be comparable. The subset relation on a collection of sets is a partial order. Total orders add comparability: for any two elements, either aRb or bRa (or both, in which case a = b). The usual ≤ on the integers is a total order.
Learning to identify which properties a relation has — and which it lacks — is the first step toward classifying it. The strategy is to test each property directly with concrete examples and counterexamples. For a finite set, you can draw the relation as a directed graph (a → b whenever aRb) and read off reflexivity (every node has a self-loop), symmetry (every edge runs in both directions), and transitivity (if a → b → c then a → c). This visual representation builds intuition rapidly and makes the abstract definitions concrete.