Relation — Definition, Examples & Properties

Relation

For example, {(1, 2), (3, 4), (1, a), (5, r)} is a relation. So is the set {(x, y): y = x²}; this is the set of all ordered pairs (x, y) for which y = x².

Ellipse centred at (2, 1) with equation (x − 2)²/9 + (y − 1)²/4 = 1. Four points marked: (−1, 1), (2, 3), (5, 1), (2, −1).

See also

Function, graph

Key Formula

R = \{(x, y) \mid x \in A \text{ and } y \in B \text{ and the pairing rule holds}\}

Where:

$R$ = The relation — a set of ordered pairs
$x$ = An element from set A (the first component of each pair)
$y$ = An element from set B (the second component of each pair)
$A$ = The set from which first components are drawn (the domain)
$B$ = The set from which second components are drawn (the range lives inside B)

Worked Example

Problem: Given the relation R = {(1, 3), (2, 6), (3, 9), (4, 12)}, find the domain, range, and determine whether R is a function.

Step 1: List all first components (x-values) of the ordered pairs to find the domain.

\text{Domain} = \{1, 2, 3, 4\}

Step 2: List all second components (y-values) to find the range.

\text{Range} = \{3, 6, 9, 12\}

Step 3: Check whether any x-value appears more than once with different y-values. Each x-value (1, 2, 3, 4) appears exactly once.

Step 4: Because every x-value is paired with exactly one y-value, this relation is also a function. The rule connecting them is

y = 3x

y = 3x

Answer: Domain = {1, 2, 3, 4}, Range = {3, 6, 9, 12}. The relation is a function described by y = 3x.

Another Example

This example shows a relation that is NOT a function, highlighting the key distinction: a relation can pair one x-value with multiple y-values, but a function cannot.

Problem: Given the relation S = {(2, 5), (3, 7), (2, 9), (4, 7)}, find the domain and range, and determine whether S is a function.

Step 1: Identify the domain by collecting all first components.

\text{Domain} = \{2, 3, 4\}

Step 2: Identify the range by collecting all second components.

\text{Range} = \{5, 7, 9\}

Step 3: Check for repeated x-values. The value x = 2 appears twice: in (2, 5) and (2, 9). It maps to two different y-values.

Step 4: Because x = 2 is paired with both 5 and 9, this relation is NOT a function. A function requires each input to have exactly one output.

Answer: Domain = {2, 3, 4}, Range = {5, 7, 9}. The relation S is not a function because the input 2 maps to two different outputs.

Frequently Asked Questions

What is the difference between a relation and a function?

Every function is a relation, but not every relation is a function. A function is a special type of relation where each input (x-value) is paired with exactly one output (y-value). If any x-value maps to two or more different y-values, the relation is not a function.

How do you tell if a relation is a function from a graph?

Use the vertical line test. Draw (or imagine) vertical lines across the graph. If every vertical line crosses the graph at most once, the relation is a function. If any vertical line crosses the graph at two or more points, the relation is not a function because that x-value has multiple y-values.

Can a relation have repeated y-values?

Yes. A relation (and even a function) can have the same y-value appear with different x-values. For example, {(1, 4), (3, 4)} is a perfectly valid function. What matters for the function test is whether any single x-value maps to more than one y-value — repeated y-values are completely fine.

Relation vs. Function

	Relation	Function
Definition	Any set of ordered pairs	A relation where each x-value maps to exactly one y-value
Repeated x-values with different y-values	Allowed	Not allowed
Vertical line test	May fail	Always passes
Example	{(1, 2), (1, 5), (3, 4)}	{(1, 2), (3, 4), (5, 6)}
Relationship	Broader category	Special case of a relation

Why It Matters

Relations are one of the first topics in algebra and precalculus that bridge the gap between sets and functions. Understanding relations is essential before you can grasp what makes a function special, which is foundational for graphing, calculus, and modeling real-world data. You will also encounter relations in coordinate geometry every time you plot points or analyze whether a graph represents a function.

Common Mistakes

Mistake: Thinking that every relation must be a function.

Correction: A relation is any set of ordered pairs — it has no restriction on how x-values pair with y-values. Only when each x maps to exactly one y does the relation qualify as a function.

Mistake: Confusing the domain and range by mixing up which component is which.

Correction: The domain is always the set of first components (x-values), and the range is always the set of second components (y-values). Remember: in an ordered pair (x, y), the input comes first.

Related Terms

Function — A relation where each input has one output
Set — A relation is defined as a set of ordered pairs
Coordinates — Ordered pairs that locate points on a plane
Graph of an Equation or Inequality — Visual representation of a relation on a coordinate plane
Domain — The set of all first components in a relation
Range — The set of all second components in a relation
Ordered Pair — The building block of every relation