Definition and Representation of Relations

Given two sets A and B, their Cartesian product A × B is the set of all possible ordered pairs (a, b) where a ∈ A and b ∈ B. For example, if A = {1, 2} and B = {x, y, z}, then A × B = {(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)}. Note that order matters: (1, x) ≠ (x, 1). The size of the Cartesian product is |A| × |B|; here 2 × 3 = 6 pairs.

A relation R from set A to set B is any subset of A × B. We write a R b (read 'a is related to b') when (a, b) ∈ R. For instance, with A = {1,2,3} and B = {1,4,9}, the relation R = {(1,1),(2,4),(3,9)} pairs each number with its square. There is no requirement that every element of A must appear, and one element of A may pair with multiple elements of B — that is what distinguishes a general relation from a function.

For a relation R from A to B: the domain is the set of all first elements (inputs) that actually appear in R; the codomain is the full set B (all possible outputs); and the range (or image) is the set of second elements (outputs) that actually appear in R. The range is always a subset of the codomain. Example: if R = {(1,2),(1,3),(2,5)}, the domain = {1,2}, the range = {2,3,5}. If B = {1,2,3,4,5}, the codomain = {1,2,3,4,5} but range ≠ codomain since 1 and 4 do not appear.

A relation can be written in roster form by listing all ordered pairs: R = {(1,3),(2,6),(3,9)}. Alternatively, set-builder notation describes the rule: R = {(x,y) | y = 3x, x ∈ {1,2,3}}. Set-builder notation is especially useful when the relation is infinite, such as R = {(x,y) | y ≥ x, x,y ∈ ℝ}, which describes the half-plane on or above the line y = x.

A mapping diagram uses two ovals — one for the domain elements and one for the codomain elements — with arrows showing which elements are paired. An element in the domain oval can have one arrow, multiple arrows, or no arrow at all. If every element in the domain oval has exactly one arrow, and no domain element has two arrows pointing to different codomain elements, the relation is a function. Mapping diagrams make it easy to check this condition visually.

When A and B are subsets of ℝ, a relation can be graphed on the coordinate plane. Each ordered pair (x, y) ∈ R corresponds to a point. A discrete relation such as {(1,2),(2,4),(3,6)} gives isolated points. A continuous relation like R = {(x,y) | y = x²} gives a smooth parabola. The vertical line test for functions emerges naturally here: if any vertical line crosses the graph at more than one point, then some x-value is paired with more than one y-value, meaning the relation is not a function.

• R — Relation (none): The relation itself — a set of ordered pairs representing the correspondence between elements of A and elements of B.
• A — Domain set (none): The set from which the first elements (inputs) of the ordered pairs are chosen.
• B — Codomain set (none): The set from which the second elements (outputs) of the ordered pairs are chosen.
• A \times B — Cartesian product (none): The set of all possible ordered pairs (a, b) with a from A and b from B. R is a subset of this larger set.
• (a,b) — Ordered pair (none): A pairing of an element a from A with an element b from B. Order matters: (a,b) ≠ (b,a) in general.

Problem. Let A = {2, 3, 4, 5} and B = {4, 6, 8, 10, 25}. Define the relation R from A to B by the rule: (x, y) ∈ R if y = 2x or y = x². (a) List R as a set of ordered pairs. (b) State the domain and range of R. (c) Is R a function? Explain.

Answer. R = {(2,4),(3,6),(4,8),(5,10),(5,25)}; domain = {2,3,4,5}; range = {4,6,8,10,25}. R is not a function because the element 5 in the domain is paired with two different elements (10 and 25) in B.

• Definition of a Function — A function is a special type of relation in which every element of the domain is paired with exactly one element of the codomain. Understanding the general definition of a relation is the essential prerequisite for recognising what makes a function more restrictive.
• Domain and Range — The concepts of domain, codomain, and range introduced here are explored in much greater depth when studying functions, especially when determining the domain and range from equations, graphs, and real-world contexts.
• The Vertical Line Test — The vertical line test is the graphical method for determining whether a relation represented on the coordinate plane is a function. It builds directly on the graphical representation of relations introduced in this article.
• Cartesian Product of Sets — A relation is formally defined as a subset of a Cartesian product. Studying the Cartesian product in depth — including how to compute it and understand its size — reinforces the set-theoretic foundation of relations.