Relations and Functions Class 12 Notes: CBSE Maths Chapter 1

Relation

• Relations in Maths is one of the very important topics for the set theory.

• Relations and functions generally tell us about the different operations performed on the sets.

• Relation in Maths can be put into term as a connection between the elements of two or more sets and the sets must be non-empty.

• A relation namely R is formed by a Cartesian product of subsets.

• It defines the relationship between two sets of values, let say from set A to set B.

• Set A is then called domain and set B is then called codomain. If $\left( a,b \right)\in R$, it shows that $a$ is related to $b$ under the relation $R$

Types of Relations:

1. Empty Relation: 

• In this there is no relation between any element of a set. 

• It is also known as void relation 

• For example: if set A is $\left\{ 2,4,6 \right\}$ then an empty relation can be $R=\left\{ x,y \right\}$where $x+y>11$ 

2. Universal Relation:

• In this each element of a set is related to every element of that set.

• For example: if set A is $\left\{ 2,4,6 \right\}$ then a universal relation can be $R=\left\{ x,y \right\}$where $x+y>0$

3. Trivial Relation: Empty relation and universal relation is sometimes called trivial relation.

4. Reflexive Relation: 

• In this each element of set (say) A is related to itself i.e., a relation R in set A is called reflexive if \[\left( a,a \right)\in R\] for every $a\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,1 \right),\left( 1,2 \right),\left( 2,2 \right),\left( 2,1 \right),\left( 3,3 \right) \right\}$ is reflexive since each element of set A is related to itself.

5. Symmetric Relation:

• A relation R in set A is called symmetric if \[\left( a,b \right)\in R\]and $\left( b,a \right)\in R$for every $a,b\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,2 \right),\left( 2,1 \right),\left( 2,3 \right),\left( 3,2 \right),\left( 3,1 \right),\left( 1,3 \right) \right\}$ is symmetric.

6. Transitive Relation:

• A relation R in set A is called transitive if \[\left( a,b \right)\in R\]and $\left( b,c \right)\in R$then $\left( a,c \right)$ also belongs to R for every $a,b,c\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,2 \right),\left( 2,3 \right),\left( 1,3 \right)\left( 2,3 \right),\left( 3,2 \right),\left( 2,2 \right) \right\}$ is transitive.

7. Equivalence Relation:

• A relation $R$ on a set A is equivalent if $R$ is reflexive, symmetric and transitive.

• For example:$R=\left\{ \left( {{L}_{1}},{{L}_{2}} \right):line{{L}_{1}}is parallel line{{L}_{2}} \right\}$,

This relation is reflexive because every line is parallel to itself

Symmetric because if ${{L}_{1}}$ parallel to ${{L}_{2}}$ then ${{L}_{2}}$ is also parallel to ${{L}_{1}}$

Transitive because if ${{L}_{1}}$ parallel to ${{L}_{2}}$ and ${{L}_{2}}$ parallel to ${{L}_{3}}$ then ${{L}_{1}}$ is also parallel to ${{L}_{3}}$

Functions

A function can have the same range mapped as that of in relation, such that a set of inputs is related to exactly one output. A function f from a set A to a set B is a rule which associates each element of set A to a unique element of set B.

• Set A is domain and set B is codomain of the function 

• Range is the set of all possible resulting values given by the function.

For example: ${{x}^{2}}$ is a function where values of $x$ will be the domain and value given by ${{x}^{2}}$ is the range.

Types of Function:

1. One-One Function: 

• A function f from set A to set B is called one-one function if no two distinct elements of A have the same image in B.

• Mathematically, a function f from set A to set B if $f\left( x \right)=f\left( y \right)$ implies that $x=y$ for all $x,y\in A$.

• One-one function is also called an injective function.

• For example: If a function f from a set of real numbers to a set of real numbers, then $f\left( x \right)=2x$ is one-one function.

2. Onto Function:

• A function f from set A to set B is called onto function if each element of set B has a preimage in set A or range of function f is equal to the codomain i.e., set B.

• Onto function is also called surjective function.

• For example: If a function f from a set of natural numbers to a set of natural numbers, then $f\left( x \right)=x-1$ is onto the function.

3. Bijective Function:

• A function f from set A to set B is called a bijective function if it is both one-one function and onto function.

• For example: If a function f from a set of real numbers to a set of real numbers, then $f\left( x \right)=2x$ is one-one function and onto function.

Composition of Function and Invertible Function

• Composition of function: Let $f:A\to B$ and $g:B\to C$ then the composite of $g$ and $f$, written as $g\circ f$ is a function from A to C such that $\left( g\circ f \right)\left( a \right)=g\left( f\left( a \right) \right)$ for all $a\in A$. (Not in the current syllabus)

• Properties of composition of function: Let $f:A\to B$, $g:B\to C$ and $h:C\to A$ then

• Composition is associative i.e., $h\left( gf \right)=\left( hg \right)f$ 

• If f and g are one-one then $g\circ f$ is also one-one

• If f and g are onto then $g\circ f$ is also onto

• Invertible function: If f is bijective then there is a function ${{f}^{-1}}:B\to A$ such that $\left( {{f}^{-1}}f \right)\left( a \right)=a$ for all $a\in A$ and $\left( {{f}^{-1}}f \right)\left( b \right)=b$ for all $b\in B$

${{f}^{-1}}$  is the inverse of the function f and is always unique. (Not in the current syllabus)

Binary Operations

• A binary operation are mathematical operations such as addition, subtraction, multiplication and division performed between two operands.

• A binary operation on a set A is defined as operations performed between two elements of set A and the result also belongs to set A. Then set A is called binary composition.

• It is denoted by $*$

For example: Binary addition of real numbers is a binary composition since by adding two real numbers the result will always be a real number.

Properties of Binary Composition:

• A binary operation $*$ on the set X is commutative, i.e., $a*b=b*a$, for every $a,b\in X$ 

• A binary operation $*$ on the set X is associative, i.e., \[a*\left( b*c \right)=\left( a*b \right)*c\], for every $a,b,c\in X$

• There exists identity for the binary operation $*:A\times A\to A$, i.e., $a*e=e*a=a$ for all $a,e\in A$

A binary operation $*:A\times A\to A$ is said to be invertible with respect to the operation $*$ if there exist an element $b$ in $A$ such that $a*b=b*a=e$, $e$ is identity element in $A$ then $b$ is the inverse of $a$ and is denoted by ${{a}^{-1}}$

Relations and Functions Class 12 Notes Mathematics

All the topics and subtopics which are covered in Relations and Functions for Class 12 are given below:

• Introduction

• Types of Relations

• Types of Functions

• Composition of functions and invertible functions

• Binary operations

5 Important Topics of Science Class 12 Chapter 1 You Shouldn’t Miss!

Importance of Science Chapter 1 Notes Relations and Functions Class 12

• The notes help clarify important ideas like relations, functions, domain, and range, making it easier to understand complex topics.

• They break down tough concepts into simpler parts, making learning more accessible and easier to manage.

• The notes include practice problems that help reinforce what you’ve learned and improve problem-solving skills.

• They offer clear summaries and key points, making it easier to review and remember important information before exams.

• By covering all the important parts of the chapter, these notes ensure thorough preparation, helping students feel more confident and ready for their exams.

Tips for Learning the Class 12 Science Chapter 1 Relations and Functions

• Start by grasping key terms like relations, functions, domain, and range to build a strong foundation.

• Draw diagrams and graphs to visualize functions and relations, helping to understand how different elements are connected.

• Work on a variety of practice problems to apply concepts and reinforce understanding, improving problem-solving skills.

• Make summary notes of important definitions and formulas to review and easily recall information during exams.

• Explain the concepts to a friend or family member to solidify your own understanding and identify any gaps in knowledge.

Conclusion

The Revision Notes for Class 12 Maths Chapter on Relations and Functions make complex ideas simpler. They explain key concepts such as relations, functions, domain, and range clearly. The notes provide helpful summaries and practice problems to reinforce learning. This chapter shows how different functions work and how to solve problems involving them. By using these notes, students can quickly review important points, improve their understanding, and feel more confident about their exam preparation. Regularly going through these notes will help students master the topic and perform better in their Class 12 Maths exams.

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