# Relations vs. Functions — What's the Difference?

Edited by Tayyaba Rehman — By Fiza Rafique — Published on January 12, 2024
Relations are any associations between sets of elements, while functions are specific types of relations where each input is associated with exactly one output.

## Key Differences

Relations in mathematics refer to any relationship between sets of elements. Functions, a subset of relations, are defined as relations where every input has a unique output.
While all functions are relations, not all relations qualify as functions. Functions require a single output for each input, whereas relations have no such restriction.
A relation can pair an element with multiple elements, but in a function, this one-to-many mapping is not allowed. Functions are, thus, more specific than general relations.
Representing relations can be done in various ways, including graphs or sets of ordered pairs. Functions, due to their specific nature, are often represented by graphs where each x-coordinate is paired with one y-coordinate.
In the context of functions, the concept of domain and range is critical, with each input (domain) associated with one output (range). In relations, the domain and range can have more complex associations.

## Comparison Chart

### Definition

Any association between sets of elements
Specific relations with one output per input

### Mapping

Can be one-to-many
Strictly one-to-one

### Graph Representation

Can have multiple y-values for one x-value
Each x-value has a single y-value

### Types

More general, various types
Subset of relations with specific properties

### Domain and Range

Can have complex associations
Each domain element has one range element

## Compare with Definitions

#### Relations

Any association between sets of elements.
In their research, they explored the relations between temperature and gas volume.

#### Functions

Specific relations with one output per input.
The function f(x) = x^2 showed a clear relationship between x and y.

#### Relations

Represented in various ways including graphs.
The graph showed the complex relations between the variables.

#### Functions

Each x-value has a single y-value in graphs.
In the function's graph, each x-coordinate corresponded to one y-coordinate.

#### Relations

Can have complex domain and range associations.
Their study on relations revealed diverse domain and range interactions.

#### Functions

Each domain element has one range element.
The function defined a clear output for every input value.

#### Relations

Can pair an element with multiple elements.
The relation between students and their preferred subjects was not exclusive.

#### Functions

Strictly one-to-one mapping.
For every input in the function, there was a unique output.

#### Relations

More general term in mathematics.
Understanding relations is fundamental before delving into specific types.

#### Functions

Subset of relations with specific properties.
Functions, a special kind of relations, have distinct mathematical properties.

#### Relations

A logical or natural association between two or more things; relevance of one to another; connection
The relation between smoking and heart disease.

#### Functions

A person's role or occupation
In my function as chief editor.

#### Relations

The connection of people by blood or marriage; kinship.

#### Functions

(Biology) The physiological activity of an organ or body part
The heart's function is to pump blood.

#### Relations

A person connected to another by blood or marriage; a relative.

#### Functions

(Computers) A procedure within an application.

#### Relations

The way in which one person or thing is connected with another
The relation of parent to child.

#### Functions

An official ceremony or a formal social occasion
Disliked attending receptions and other company functions.

#### Relations

The mutual dealings or connections of persons, groups, or nations in social, business, or diplomatic matters
International relations.

#### Functions

Something closely related to another thing and dependent on it for its existence, value, or significance
Growth is a function of nutrition.

#### Relations

Sexual intercourse.

#### Functions

A variable so related to another that for each value assumed by one there is a value determined for the other.

#### Relations

The act of telling or narrating.

#### Functions

A rule of correspondence between two sets such that there is exactly one element in the second set assigned to each element in the first set. Also called mapping.

#### Relations

A narrative; an account.

#### Functions

To have or perform a function; serve

#### Relations

(Mathematics) A correspondence between two sets, consisting of a set of ordered pairs, the first element of each of which is from the first set, and the second element of each of which is from the second set. If A = {1,2} and B = {3,4}, then {(1,3), (1,4)} is a relation from A to B.

#### Functions

To deal with or overcome the challenges of everyday life
For weeks after his friend's funeral he simply could not function.

#### Relations

(Law) The principle by which an action done on a certain date is treated as having been done on an earlier date. Also called relation back.

#### Functions

Plural of function

#### Relations

Plural of relation

#### Relations

Mutual dealings or connections or communications among persons or groups

## Common Curiosities

#### What’s an example of a relation that’s not a function?

A relation where an input is associated with multiple outputs.

#### How are functions represented graphically?

By graphs where each x-value is paired with one y-value.

#### Can functions have multiple y-values for one x-value?

No, functions must have a single output for each input.

#### What are relations in mathematics?

Associations between sets of elements.

#### What is the domain of a function?

The set of all possible input values.

#### What defines a function?

A relation where each input has exactly one output.

#### Can functions have complex domain and range associations?

Functions have simpler associations, with one domain element linked to one range element.

#### How do relations and functions differ in mapping?

Relations can have one-to-many mapping, while functions cannot.

#### Can a relation be a function?

Yes, if it has a unique output for every input.

#### Are all functions relations?

Yes, all functions are a type of relation.

#### What is the range of a function?

The set of all possible output values.

#### Is understanding relations important for learning functions?

Yes, it provides a foundational understanding.

#### Are there different types of functions?

Yes, there are various types like linear, quadratic, and exponential functions.

#### How do relations contribute to mathematical studies?

They provide a broader understanding of element associations.

#### Why are functions important in mathematics?

They are fundamental in understanding systematic relationships between variables.