Relation: Definition

Definition

A relation from a set $A$ to a set $B$ is any subset of the Cartesian product $A \times B$ .

If $A$ and $B$ are sets, then:

$R \subseteq A \times B$

Each element of $R$ is an ordered pair $(a,b)$ , where $a \in A$ and $b \in B$ .

If the relation is defined on a single set $A$ , then it is called a binary relation on $A$ , and it is a subset of:

$A \times A$

Example

Let $A = \{1,2,3\}$ and $B = \{a,b\}$ .

Then:

$A \times B = \{(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)\}$

A possible relation $R$ from $A$ to $B$ is:

$R = \{(1,a),(2,b),(3,a)\}$

This means:

$1$ is related to $a$
$2$ is related to $b$
$3$ is related to $a$

A relation can be represented in many ways, such as:

roster form
set-builder form
arrow diagram
matrix form
graph form

Main Content

1. Cartesian Product and Ordered Pairs

The Cartesian product of two sets is the set of all possible ordered pairs formed by taking one element from the first set and one from the second set.
Relations are built from Cartesian products, so understanding ordered pairs is the first step in understanding relations.

If:

$A = \{1,2\}, \quad B = \{x,y\}$

then:

$A \times B = \{(1,x),(1,y),(2,x),(2,y)\}$

Any subset of this product is a relation from $A$ to $B$ .

Important points:

Order matters in ordered pairs: $(a,b) \neq (b,a)$ in general.
A relation may contain one pair, many pairs, or even no pairs at all.
The empty relation is a valid relation because the empty set is a subset of every set.

Example

If:

$R = \{(1,x),(2,y)\}$

then $R$ is a relation from $A$ to $B$ . It is not necessary to include all pairs from $A \times B$ .

2. Types of Relations

Relations are classified based on the sets involved and the properties they satisfy.

a) Relation from one set to another set

A relation from $A$ to $B$ consists of pairs where the first element comes from $A$ and the second from $B$ .

Example:

Students $\to$ courses they take
Employees $\to$ departments they work in

b) Relation on a set

A relation on a set $A$ is a relation from $A$ to itself.

Example:

“less than or equal to” on integers
“is equal to” on numbers
“is friends with” among people

c) Universal relation

The universal relation on $A$ is:

$A \times A$

Every element is related to every element.

d) Empty relation

The empty relation is:

$\varnothing$

No element is related to any element.

Important points:

Relations can be finite or infinite.
Relations may be symmetric, transitive, reflexive, etc., depending on their structure.
Relations on sets are especially important in abstract algebra and discrete mathematics.

Example

Let $A = \{1,2,3\}$ .

A relation on $A$ could be:

$R = \{(1,1),(2,2),(1,2)\}$

This is a relation on $A$ because every first and second element belongs to $A$ .

3. Representation of Relations

Relations can be described in several ways, and each representation is useful in different situations.

a) Roster form

The relation is written as a set of ordered pairs.

Example:

$R = \{(1,2),(2,3),(3,4)\}$

b) Set-builder form

The relation is described by a property satisfied by the ordered pairs.

Example:

$R = \{(x,y) \mid x < y,\ x,y \in A\}$

c) Arrow diagram

Elements of one set are shown with arrows pointing to related elements in another set.

For example, if $A = \{1,2,3\}$ and $B = \{a,b\}$ , and

$R = \{(1,a),(2,b),(3,a)\}$

then the arrows are:

1  ---> a
2  ---> b
3  ---> a

d) Matrix representation

For finite sets, relations can be represented by a matrix containing 1s and 0s.

If $A = \{a_1,a_2,\dots,a_n\}$ , then the relation matrix $M_R$ has:

entry 1 if $(a_i,a_j)\in R$
entry 0 otherwise

e) Graph representation

A relation on a set can be represented using a directed graph:

each element is a vertex
each ordered pair is a directed edge

Example: If $(1,2)\in R$ , draw an arrow from 1 to 2.

Important points:

Roster form is simple and direct.
Set-builder form is compact and mathematical.
Arrow diagrams and graphs help visualize relations.
Matrices are useful in computation and theorem proving.

Working / Process

1. Identify the sets involved

Determine the source set $A$ and, if needed, the target set $B$ .
If the relation is on one set, then both positions in the ordered pair come from the same set.

2. Form the Cartesian product

Write all possible ordered pairs in $A \times B$ or $A \times A$ .
This gives the complete set of possible related pairs.

3. Select the required pairs according to the rule

Choose only those ordered pairs that satisfy the condition defining the relation.
The selected subset is the relation.

Example

Let $A = \{1,2,3\}$ and define the relation $R$ by:

$xRy \text{ if } x < y$

Step 1: Identify the set $A$ .

Step 2: Form $A \times A$ :

$\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$

Step 3: Keep only pairs where the first number is less than the second:

$R = \{(1,2),(1,3),(2,3)\}$

So this is the relation “less than” on the set $A$ .

Advantages / Applications

Relations help model real-life connections such as student-course enrollment, family links, social networks, and scheduling dependencies.
They are essential for defining functions, equivalence relations, and partial orders, which are central topics in higher mathematics.
Relations are used in computer science for databases, graph theory, automata, logic, and algorithm design.

Additional applications

In databases, relations are used to organize data in tables.
In graph theory, relations correspond to directed edges.
In mathematics, they help prove properties of structures and simplify reasoning in theorem proving.
In logic, relations express predicates involving more than one object.

Summary

A relation is a subset of a Cartesian product.
Relations describe how elements of one set are connected to elements of another set or within the same set.
They can be represented in several forms such as ordered pairs, arrows, matrices, and graphs.