Relations and Functions

What are Relations and Functions?

A relation in mathematics connects elements of two sets. It helps in identifying common terms between sets or functions. The total number of relations between two sets $A$ and $B$ depends on their sizes. Relations are fundamental to the study of functions, and they are widely used in areas like database systems, graph theory, and social networks.

Relation: Definition

A relation is a way to associate elements of one set with elements of another set. If two sets have values, a relation connects elements of the first set with those of the second.

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the Cartesian product $A \times B$.

The subset is formed by specifying how the first element relates to the second in ordered pairs.

If $(a,b)$ belongs to $R$, where $a \in A$ and $b \in B$, it is written as $a R b$.

Example:
Let $A = \{1, 2\}$ and $B = \{x, y\}$. A relation can be: $R = \{(1, x), (2, y)\}$.
This means $1$ is related to $x$ and $2$ is related to $y$.

Domain of a Relation

The domain of a relation is the set of all first elements in the ordered pairs of the relation. It represents all possible inputs that have at least one output.

For example, if $R = \{(1, c), (5, a), (8, b)\}$, then the domain is: $\{1, 5, 8\}$.

Range of a Relation

The range of a relation is the set of all output values or second elements in the ordered pairs.

For the relation $R = \{(1, c), (5, a), (8, b)\}$, the range is: $\{c, a, b\}$.

Function: Definition

A function is a special kind of relation where each element in set $A$ is related to exactly one element in set $B$.

Formally, if $A$ and $B$ are two non-empty sets, a function $f: A \rightarrow B$ assigns each element $x \in A$ a unique element $f(x) \in B$.

This is written as: $f: A \rightarrow B$ (read as “$f$ is mapping from $A$ to $B$”).

Every function is a relation, but not every relation is a function.

First and second images represent a function as all the elements in set $A$ is mapped a element in set $B$ while the element $d$ in set $A$ is not mapped to any element in set $B$.

A function can be said as a subset of relation.

Image of a Function

The image of a function is the set of all output values produced from its domain.

Given a function $f: A \rightarrow B$ and a subset $X \subseteq A$, the image is:
$f(X) = \{f(x) \mid x \in X\}$.

If we consider the entire domain, the image is the range:
$\text{Image}(f) = f(A) = {f(a) \mid a \in A}$.

Pre-image of a Function

The pre-image refers to the set of all inputs that map to a particular output.

For a function $f: A \rightarrow B$ and $Y \subseteq B$, the pre-image is:
$f^{-1}(Y) = \{x \in A \mid f(x) \in Y\}$.

For a single output value $y \in B$:
$f^{-1}({y}) = \{x \in A \mid f(x) = y\}$.

For example, in the ordered pair $(1,2)$, $1$ is the pre-image of $2$ under the function.

Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined.

If $f: A \rightarrow B$, then $A$ is the domain i.e. all the elements of set $A$ is called the Domain of the function.

Co-domain of a Function

The co-domain of a function is the set $B$ that includes all possible outputs, whether or not they are actually achieved by the function.

The range is always a subset of the co-domain and can sometimes be equal to it.

Range of a Function

The range of a function is the set of all actual output values corresponding to elements in the domain.

For example, for $f: A \rightarrow B$ where $f(x) = x^2$ and $A$ is the set of all integers, the range is:
$\{1, 4, 9, 16, \dots \}$.

Another example:
Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 4, 8, 16, 27, 64, 125\}$.
If $f(x) = x^3$, then:

• Domain = $A$

• Co-domain = $B$

• Range = $\{1, 8, 27, 64, 125\}$.

The range is always a subset of the co-domain and the Range can be equal to the co-domain in some cases.

Representation of Relation and Function

In relations and functions, it’s important to understand how they can be visually or mathematically represented. Different forms (roster and set builder form) help in identifying patterns, solving problems, and understanding how elements from one set relate to another. Below are the common ways to represent relations and functions along with examples based on the topics covered in relations and functions class 12 and relations and functions examples.

1. Roster Form

In the Roster Form, relations and functions are written as a list of ordered pairs enclosed within curly brackets. Every pair explicitly shows how elements from set $A$ are connected to elements from set $B$.

Example:
Let $A = \{1, 2\}$ and $B = \{x, y\}$.
The relation is:
$R = \{(1, x), (2, y)\}$.

This is one of the easiest ways to represent relations and functions, especially for small sets.

2. Set Builder Form

The Set Builder Form uses a rule or condition to define the relationship between elements of sets, rather than listing all pairs. The symbols ‘:’ or ‘|’ are read as ‘such that’.

Example:
Let $A = \{1,2,3,4,5\}$ and $B = \{1,2,3,4,5,6,7,8,9\}$.
Define a function $f: A \rightarrow B$ by $f(x) = x + 1$.
Then the relation is written as:
$R = \{(x, y) : x \in A, y \in B \text{ and } y = x + 1\}$.

This form is widely used in relations and functions class 12 solutions where problems involve conditions or formulas.

3. Arrow Diagram

The Arrow Diagram visually represents how elements of set $A$ map to elements of set $B$. Arrows connect each element from the domain to its corresponding image in the codomain.

Example:
Let $A = \{1,2,3,4,5\}$ and $B = \{1,4,8,16,27,64,125\}$.
Define $f: A \rightarrow B$ as $f(x) = x^3$.
The diagram shows arrows mapping $1 \rightarrow 1$, $2 \rightarrow 8$, $3 \rightarrow 27$, and so on.

Arrow diagrams are useful when solving relations and functions class 12 important questions, making abstract concepts easier to understand.

4. Tabular Form

The Tabular Form is another way to represent relations and functions, where the elements of the domain and codomain are arranged in a table format. This makes it easier to visualize how each input corresponds to an output.

Example:
Let $A = \{1,2,3,4,5\}$ and $B = \{1,4,8,16,27,64,125\}$.
Define the function $f: A \rightarrow B$ by $f(x) = x^3$.
The table representing this function is:

Difference Between Relations and Functions

Types of Relations and Functions

In mathematics, relations and functions can be classified into various types based on the properties they possess. Understanding these types is important for solving problems in relations and functions class 12, relations and functions class 11, and for preparing for exams like JEE Main mathematics relations and functions.

1. Empty Relation

A relation $R$ on a set $A$ is called an empty relation if no element of $A$ is related to any element of $A$, i.e., $R = \emptyset$.

Example:
Let $A = \{2, 4, 6\}$ and $R = \{(a, b) : a, b \in A$ and $a + b$ is odd$\}$.
Here, $R$ contains no element, so it is an empty relation on $A$.

2. Universal Relation

A relation $R$ on a set $A$ is a universal relation if every element of $A$ is related to every other element of $A$, i.e., $R = A \times A$.

Example 1:
Let $A = \{2,4\}$ and $R = \{(2,2), (2,4), (4,2), (4,4)\}$.
Since $R = A \times A$, it is a universal relation.

Example 2:
Let $A = \{1,2,3\}$ and $R = \{(a, b) : |a - b| > -2, a, b \in A\}$.
Each ordered pair satisfies this condition, so $R$ is a universal relation.

3. Identity Relation

A relation where every element of $A$ is related to itself only is called an identity relation, denoted by $I_A$.

It is defined as: $R = \{(a, b) : a \in A, b \in A$ and $a = b\}$ or $I_A = \{(a, a) : a \in A\}$.

Example:
Let $A = {2,4,6}$. Then, $I_A = \{(2,2), (4,4), (6,6)\}$.

4. Reflexive Relation

A relation $R$ on set $A$ is reflexive if $(a, a) \in R$ for every $a \in A$.

Example:
Let $A = \{1,2,3\}$

• $R_1 = \{(1,1), (2,2), (3,3)\}$

• $R_2 = \{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3)\}$
  Both are reflexive relations, but

• $R_3 = \{(1,1), (2,2), (2,3), (3,2)\}$ is not reflexive because $(3,3)$ is missing.

5. Symmetric Relation

A relation $R$ is symmetric if whenever $a R b$ holds, then $b R a$ also holds for all $a, b \in A$.

Example:
Let $A = \{1,2,3\}$

• $R_1 = \{(1,2), (2,1)\}$

• $R_2 = \{(1,2), (2,1), (1,3), (3,1)\}$ are symmetric, while

• $R_3 = \{(2,3), (1,3), (3,1)\}$ is not symmetric as $(3,2)$ is missing.

6. Transitive Relation

A relation $R$ is transitive if whenever $a R b$ and $b R c$ hold, then $a R c$ also holds for all $a, b, c \in A$.

Example:
Let $A = \{1,2,3\}$

• $R_1 = \{(1,2), (2,3), (1,3), (3,2)\}$ is not transitive.

• $R_2 = \{(2,3), (3,1)\}$ is not transitive.

• $R_3 = \{(1,3), (3,2), (1,2)\}$ is transitive.

7. Equivalence Relation

A relation is an equivalence relation if it is reflexive, symmetric, and transitive.

Example:
Let $R = \{(1,1), (1,2), (2,1), (2,2)\}$.
Since $R$ satisfies all three properties, it is an equivalence relation.

8. Inverse Relation

An inverse relation swaps each ordered pair from a relation.

If $R \subseteq A \times B$, then
$R^{-1} = \{(b, a) : (a, b) \in R\}$.

Example:
Let $R = \{(1,2), (3,4)\}$.
Then, $R^{-1} = \{(2,1), (4,3)\}$.

Types of Functions

In mathematics, functions are classified into various types based on how elements from the domain relate to elements in the codomain. These types are essential for understanding concepts in relations and functions class 12, relations and functions examples, and solving problems in class 12 relations and functions important questions.

1. One-to-One Function (Injective)

A one-to-one function, also known as an injective function, is a function where every element in the domain has a unique image in the codomain. In other words, distinct elements of set $A$ are mapped to distinct elements in set $B$.

A function $f: X \rightarrow Y$ is called injective if different elements of $X$ always map to different elements of $Y$. That is, no two elements of set $X$ can have the same image.

Example:
Let $f: X \rightarrow Y$ be defined by $f(x) = x$, where
$X = \{-2, 2, 4, 6\}$ and $Y = \{-2, 2, 4, 6\}$.
Here, every element of $X$ has a distinct image in $Y$, so $f$ is a one-to-one function.

2. Many-to-One Function

A many-to-one function is a type of function where two or more elements from the domain are mapped to the same element in the codomain. This is a common scenario in problems related to relations and functions class 12, relations and functions examples, and relations and functions class 12 solutions.

A function $f: X \rightarrow Y$ is called many-to-one if two or more elements of set $X$ have the same image in set $Y$. That is, it is not a one-to-one function.

In other words, if $f: X \rightarrow Y$ is not injective, it is a many-to-one function.

Example:
In the case below, two elements $x_2$ and $x_3$ correspond to the same image $y_3$:
$f(x_2) = f(x_3) = y_3$.

A well-known example is the function:
$f(x) = x^2$
Domain: All real numbers $(\mathbb{R})$
Codomain: Non-negative real numbers $[0, \infty)$

Here, both $x$ and $-x$ have the same image, so it’s a many-to-one function.

3. Onto Function (Surjective)

A function $f: X \rightarrow Y$ is an onto function (or surjective) if every element of set $Y$ is the image of at least one element from set $X$. That is, Range = Codomain.

Example:
Let $X = \{x_1, x_2, x_3, x_4\}$ and $Y = \{y_1, y_2, y_3\}$,
then a function $f: X \rightarrow Y$ can be onto if every element in $Y$ is mapped to by some element from $X$.

4. Bijective Function

A function is bijective if it is both one-to-one and onto. Such functions create perfect pairings between the domain and codomain.

Example:
Let $X_1 = \{1, 2, 3\}$ and $X_2 = \{x, y, z\}$
A function $f: X_1 \rightarrow X_2$ is bijective if every element of $X_1$ maps uniquely to every element of $X_2$.

5. Odd and Even Functions

Let's describe about even and odd functions:

• A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$. Odd functions are symmetric about the origin.

• A function is even if $f(-x) = f(x)$ for all $x$. Even functions are symmetric about the $y$-axis.

6. Identity Function

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x$ for all $x$ is called the identity function.

Domain: $\mathbb{R}$
Range: $\mathbb{R}$

7. Constant Function

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = C$ where $C$ is a constant is a constant function.

Domain: $\mathbb{R}$
Range: ${C}$

8. Polynomial Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$f(x) = a_0 + a_1x + \dots + a_nx^n$
where $n \in \mathbb{N}$ and $a_0, a_1, ..., a_n \in \mathbb{R}$ is called a polynomial function.

9. Composite Function

Given functions $f: A \rightarrow B$ and $g: B \rightarrow C$, the composition $g \circ f$ is defined as: $g \circ f(x) = g(f(x))$ for all $x \in A$.

10. Rational Function

A rational function is of the form $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomial functions and $g(x) \neq 0$.

Example:
$f(x) = \frac{x+1}{x+2}$, defined for $x \in \mathbb{R} - {-2}$.

11. Modulus Function

The modulus function is defined as:
$f(x) = |x| =
\begin{cases}
x, & x \geq 0 \\
-x, & x < 0
\end{cases}$
for all $x \in \mathbb{R}$.
Domain: $\mathbb{R}$
Range: $\mathbb{R}^{+} \cup {0}$

12. Signum Function

The signum function is defined as:
$f(x) =
\begin{cases}
1, & x > 0 \\
0, & x = 0 \\
-1, & x < 0
\end{cases}$
Domain: $\mathbb{R}$
Range: ${1, 0, -1}$

13. Greatest Integer Function

The greatest integer function is defined by $f(x) = [x]$, where $[x]$ is the largest integer less than or equal to $x$.

Example:
$f(x) = [x] = -1$ for $-1 \leq x < 0$,
$f(x) = [x] = 0$ for $0 \leq x < 1$,
and so on.

14. Periodic Function

A function $f(x)$ is called periodic if there exists a positive real number $T$ such that $f(x+T) = f(x)$ for all $x$ in the domain. The smallest such $T$ is called the period.

Relations and Functions: Formulae and Operations

List of Topics related to relations and functions according to NCERT/JEE MAIN

This section covers the list of topics related to relations and functions according to NCERT and JEE Main syllabus. It helps students focus on important concepts, examples, and formulas needed for class 11 and 12 preparation.

Recommended Books for Relations and Functions

This section provides a list of recommended books for relations and functions that are useful for class 11 and 12 students. These books cover concepts, examples, and practice questions aligned with NCERT and JEE Main syllabus.

NCERT Resources

In this section, you’ll find NCERT resources for Relations and Functions, including class 12 Maths notes, solutions, and exemplar problems. These are helpful for understanding concepts and practicing questions as per the NCERT syllabus.

NCERT Maths Notes for Class 12th Chapter 1 - Relations and Functions

NCERT Maths Solutions for Class 12th Chapter 1 - Relations and Functions

NCERT Maths Exemplar Solutions for Class 12th Chapter 1 - Relations and Functions

NCERT Subjectwise Resources

Explore NCERT subjectwise resources for Relations and Functions, including notes, solutions, and exemplar problems. These materials are designed to help class 12 students strengthen their understanding and prepare effectively for exams like JEE Main.

Practice Questions Based on Relations and Functions

Practice questions based on Relations and Functions help students apply concepts, improve problem-solving skills, and master important topics for class 12 exams and competitive tests like JEE Main.