A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A function f: A → B is a relation where each element of A (the domain) maps to exactly one element of B (the codomain). The range is the set of all elements in B that are actually outputs of f. Distinguishing codomain from range is critical: the codomain is fixed by definition, while the range is determined by which outputs are attained.
You already know that a Cartesian product A × B consists of ordered pairs (a, b), and that a relation is a subset of A × B. A function f: A → B is a special kind of relation: one where every element of A appears as the first coordinate of exactly one pair. In other words, every input has exactly one output. What distinguishes a function from a general relation is this uniqueness requirement — no input maps to two different outputs, and no input is left without an output.
The three terms domain, codomain, and range capture different aspects of a function's scope. The domain is the set A of all permitted inputs — the function must be defined on every element of A. The codomain is the set B declared as the target — it is a promise about where outputs land, chosen as part of the function's definition. The range (also called image) is {f(a) : a ∈ A} ⊆ B — the collection of outputs actually produced. The range is always a subset of the codomain, but can be strictly smaller.
The distinction between codomain and range is subtle but critical. Consider f: ℝ → ℝ defined by f(x) = x². The codomain is ℝ (all real numbers), but the range is [0, ∞) — no output is negative. You could alternatively write f: ℝ → [0, ∞), declaring the codomain to be just the nonnegatives. These are technically different functions even though the formula is identical, because the codomain is part of the function's specification. This matters enormously for surjectivity: f is surjective (onto) if and only if every element of the codomain is achieved as an output — i.e., range = codomain. Whether f(x) = x² is surjective depends entirely on the declared codomain.
This framework also clarifies well-definedness. When you define a function by a formula or rule, you must verify: (1) the formula applies to every element of the domain, and (2) each domain element maps to a unique output. In abstract algebra, functions are often defined on equivalence classes, and checking well-definedness — verifying the output doesn't depend on which representative you chose — is a required proof step. The domain-codomain-range framework makes that checklist precise.