Functions as Relations

Set Theory · Axiom Academy

LESSON Functions: Formal Definition From relations to functions: the special property of unique outputs 1. Function as a Special Relation A function from set A to set B, written f: A → B, is a relation f ⊆ A × B with a crucial property: Unique output property: For each a ∈ A, there is exactly one b ∈ B such that (a, b) ∈ f. In symbols: ∀a ∈ A, ∃!b ∈ B: (a, b) ∈ f We write f(a) = b to denote the unique element b associated with a. The key difference from arbitrary relations: each input determines exactly one output, no ambiguity allowed! 2. Notation: Domain, Codomain, Range Domain: The set A of all possible inputs Codomain: The set B where outputs live (target set) Range (Image): The subset f(a) : a ∈ A ⊆ B of actual outputs Important distinction: The codomain is where outputs can be, the range is where they actually are . Example: f: ℕ → ℕ defined by f(n) = 2n has domain ℕ, codomain ℕ, but range = 0, 2, 4, 6, ... (even numbers). 3. Function vs Arbitrary Relation Not every relation is a function. Let's see what can go wrong: Valid function: Each input maps to exactly one output Not a function (multiple outputs): Some input maps to two or more outputs Not a function (missing outputs): Some element of domain has no output Example of non-function: The relation R = (1,2), (1,3), (2,4) on 1,2 × 2,3,4 Why? The element 1 relates to both 2 and 3 (violates unique output) The graph of a function f: A → B is the subset of A × B containing all ordered pairs (a, f(a)):

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