Introduction to Relations

Learn about Cartesian products, relations, domain, co-domain, and range for JEE Mathematics.

5 min read

Introduction

A relation describes a connection between elements of two sets. Before understanding relations, we need to understand the Cartesian product.

Cartesian Product

Definition

The Cartesian product of sets $A$ and $B$, denoted $A \times B$, is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

$$\boxed{A \times B = \{(a, b) : a \in A \text{ and } b \in B\}}$$

Example

If $A = \{1, 2\}$ and $B = \{x, y\}$:

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

Properties

  1. Not commutative: $A \times B \neq B \times A$ (unless $A = B$ or one is empty)
  2. Cardinality: $|A \times B| = |A| \times |B|$
  3. Empty product: $A \times \emptyset = \emptyset = \emptyset \times A$

Graphical Representation

For $A = \{1, 2\}$ and $B = \{a, b\}$:

  B
  │
b │  •  •
  │
a │  •  •
  │
  └──────── A
     1  2

Each point represents an ordered pair.

Ordered Pairs

In an ordered pair $(a, b)$:

  • Order matters: $(1, 2) \neq (2, 1)$
  • $(a, b) = (c, d) \iff a = c$ AND $b = d$

What is a Relation?

Definition

A relation $R$ from set $A$ to set $B$ is a subset of $A \times B$.

$$R \subseteq A \times B$$

If $(a, b) \in R$, we write $aRb$ and say “$a$ is related to $b$”.

Example

Let $A = \{1, 2, 3\}$ and $B = \{1, 4, 9\}$.

Define relation $R$: “$a$ is the square root of $b$”

$$R = \{(1, 1), (2, 4), (3, 9)\}$$

Here, $1R1$, $2R4$, $3R9$ are true statements.

Relation on a Set

A relation $R$ from set $A$ to itself ($R \subseteq A \times A$) is called a relation on $A$.

Example: On $A = \{1, 2, 3\}$, define $R$: “$a$ divides $b$”

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

Domain, Co-domain, and Range

Definitions

For a relation $R$ from $A$ to $B$:

TermDefinitionNotation
DomainSet of first elements of ordered pairs in $R$$\text{Dom}(R)$
Co-domainThe set $B$-
RangeSet of second elements of ordered pairs in $R$$\text{Range}(R)$

Important Relationship

$$\boxed{\text{Range}(R) \subseteq \text{Co-domain}}$$

The range is always a subset of the co-domain!

Example

Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d, e\}$.

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

  • Domain = $\{1, 2, 3\}$ (first elements)
  • Co-domain = $\{a, b, c, d, e\}$ (the set $B$)
  • Range = $\{a, b, c\}$ (second elements)

Note: 4 is not in the domain, and $d, e$ are not in the range.

Common Confusion

Range ≠ Co-domain (in general)

Range only includes elements that are actually “hit” by the relation.

Ways to Represent Relations

1. Roster Form

List all ordered pairs:

$$R = \{(1, 2), (2, 4), (3, 6)\}$$

2. Set-Builder Form

Describe the relationship:

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

3. Arrow Diagram

Draw arrows from elements of $A$ to related elements of $B$:

  A          B
  1  ──────► 2
  2  ──────► 4
  3  ──────► 6

4. Matrix Representation

For $A = \{a_1, a_2, ..., a_m\}$ and $B = \{b_1, b_2, ..., b_n\}$:

Create an $m \times n$ matrix $M$ where:

$$M_{ij} = \begin{cases} 1 & \text{if } (a_i, b_j) \in R \\ 0 & \text{otherwise} \end{cases}$$

Example: For $R = \{(1, a), (2, b), (2, c)\}$ on $A = \{1, 2\}$ and $B = \{a, b, c\}$:

$$M = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}$$

Number of Relations

Counting Relations

Since a relation from $A$ to $B$ is a subset of $A \times B$:

$$\boxed{\text{Number of relations} = 2^{|A| \times |B|}}$$

Example: If $|A| = 2$ and $|B| = 3$:

  • $|A \times B| = 6$
  • Number of relations = $2^6 = 64$

Special Cases

Relation TypeCount
Empty relation1
Universal relation1
Non-empty relations$2^{mn} - 1$

Inverse of a Relation

Definition

The inverse of relation $R$ from $A$ to $B$, denoted $R^{-1}$, is:

$$R^{-1} = \{(b, a) : (a, b) \in R\}$$

$R^{-1}$ is a relation from $B$ to $A$.

Example

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

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

Properties

  1. $(R^{-1})^{-1} = R$
  2. Domain of $R^{-1}$ = Range of $R$
  3. Range of $R^{-1}$ = Domain of $R$

Interactive Demo: Visualize Relations

Explore relations, domain, range, and inverse relations graphically.

Practice Problems

  1. Find all ordered pairs in $A \times B$ where $A = \{1, 2\}$ and $B = \{3\}$.

  2. Let $R = \{(x, y) : y = x + 1\}$ on $A = \{0, 1, 2, 3\}$. List all elements of $R$.

  3. Find the domain and range of $R = \{(1, 4), (2, 5), (3, 6), (4, 6)\}$.

  4. How many relations are possible from a set of 3 elements to a set of 2 elements?

Quick Check
If $R$ is a relation from $A$ to $B$ with $|A| = 3$ and $|B| = 4$, what is the maximum number of ordered pairs in $R$?