Understanding relations and functions in mathematics is like learning to navigate a map – it helps students move from one point to another with purpose. In the Relations and Functions Class 12 NCERT Solutions, the first question that comes to students' minds is, "What are relations and functions?". The answer is simple: a relation is like a connection between a student and all the subjects in the syllabus. On the other hand, a function is like a special subject, which is the student's favourite subject. The main purpose of these NCERT Solutions for class 12, Relations and Functions, is to make learning easier for students and to explain this topic more easily.
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NCERT Solution for Class 12 Maths Chapter 1 Solutions: Download PDF
NCERT Solutions for Class 12 Maths Chapter 1: Exercise Questions
Class 12 Maths NCERT Chapter 1: Extra Question
Relations and Functions Class 12 NCERT Solutions: Topics
Class 12 Maths NCERT Chapter 1, Relations and Functions: Important Formulae
Approach to Solve Questions of Relations and Functions Class 12
What Extra Should Students Study Beyond the NCERT for JEE?
NCERT Solutions for Class 12 Maths: Chapter Wise
NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
Relations and functions are not just theory-based, but they also come in handy in real life. People can apply it during mapping, programming, and data handling. These NCERT solutions, created by Careers360 experts with years of experience in the field, provide students with quality practice questions. After checking these NCERT solutions for Class 12 Maths, students can also check the NCERT Exemplar Solutions and notes for Class 12 Maths NCERT Chapter 1, Relations and Functions, to understand this chapter better. For a structured syllabus, concise notes, and PDFs, click this NCERT link.
NCERT Solution for Class 12 Maths Chapter 1 Solutions: Download PDF
Students who wish to access the Class 12 Maths Chapter 1 Solutions PDF can click on the link below to download the complete solution in PDF format.
y and x work at the same place, i.e. (y,x)∈R, so it is symmetric.
(x,y),(y,z)∈R means x and y work at the same place, and also y and z work at the same place. It states that x and z work at the same place, i.e. (x,z)∈R. So, it is transitive.
Hence, it is reflexive, symmetric, and transitive.
(12)>(12)2
So,R is not reflexive.
Now,
(1,2)∈R because 1<4.
But, 4≮1, i.e. 4 is not less than 1
So, (2,1)∉R
Hence, it is not symmetric.
(3,2)∈R and (2,1.5)∈R as 3<4 and 2<2.25
Since (3,1.5)∉R because 3≮2.25
Hence, it is not transitive.
Thus, we can conclude that it is neither reflexive, nor symmetric, nor transitive.
(12,12)∉R because 12∉(12)3
So, it is not symmetric.
Now, (1,2)∈R because 1<23but(2,1) \notin Rbecause2 \nless 1^3Itisnotsymmetric(3,1.5) \in Rand(1.5,1.2) \in Ras3<1.5^3and1.5<1.2^3.But,(3,1.2) \notin Rbecause3 \nless 1.2^3$
So it is not transitive
Thus, it is neither reflexive, nor symmetric, nor transitive.
R={(P,Q): distance of the point P from the origin is the same as the distance of the point Q from the origin }
The distance of point P from the origin is always the same as the distance of the same point P from another origin, i.e. (P,P)∈R
∴R is reflexive.
Let (P,Q)∈R, i.e. the distance of the point P from the origin is the same as the distance of the point Q from the origin.
This is the same as the distance of point Q from the origin, same as the distance of point P from the origin, i.e. (Q,P)∈R
∴R is symmetric.
Let (P,Q)∈R and (Q,S)∈R
i.e. distance of the point P from the origin is the same as the distance of the point Q from the origin, and also the distance of the point Q from the origin is the same as the distance of the point S from the origin.
We can say that the distance of points P,Q,S from the origin is the same. This means a distance of point P from the origin is the same as the distance of point S from the origin, i.e. (P,S)∈R
∴R is transitive.
Hence, R is an equivalence relation.
The set of all points related to a point P≠(0,0) are points whose distance from the origin is the same as the distance of point P from the origin.
In other words, we can say there is a point 0(0,0) as the origin and the distance between point 0 and point P be k=OP; then the set of all points related to P is at a distance k from the origin.
Hence, this set of points forms a circle with the centre as the origin, and this circle passes through point P.
∴f is one- one i.e. injective.
For 3∈N there is no x in N such that f(x)=x2=3
∴f is not onto, i.e. not surjective.
Hence, f is injective but not surjective.
Therefore,f is not one- one i.e. not injective.
For −3∈Z there is no x in Z such that f(x)=x2=−3
∴f is not onto, i.e. not surjective.
Hence, f is neither injective nor surjective
Therefore,f is not one- one i.e. not injective.
For −3∈R there is no x in R such that f(x)=x2=−3
∴f is not onto, i.e. not surjective.
Hence, f is not injective and not surjective.
∴f is one- one i.e. injective.
For 3∈N there is no x in N such that f(x)=x3=3
∴f is not onto, i.e. not surjective.
Hence, f is injective but not surjective.
∴f is one- one i.e. injective.
For 3∈Z there is no x in Z such that f(x)=x3=3
∴f is not onto, i.e. not surjective.
Hence, f is injective but not surjective.
As we can see f(1)=f(2)=1, but 1≠2
So it is not one-one.
Now, f(x) takes only 3 values (1,0,−1) for the element -3 in codomain R, there does not exist x in domain R such that f(x)=−3.
So it is not onto.
Hence, the signum function is neither one-one nor onto.
f(2)=22=1 and f(1)=1+12=1
As we can see f(1)=f(2)=1 but 1≠2
∴f is not one-one.
Let, n∈N (N=co-domain)
case1 n be even
For r∈N,n=2r
then there is 4r∈N such that f(4r)=4r2=2r
case2 n be odd
For r∈N,n=2r+1
then there is 4r+1∈N such that f(4r+1)=4r+1+12=2r+1
∴f is onto.
f is not one-one but onto
Hence, the function f is not bijective
∴f(a)=f(b) does not imply that a=b
example : f(2)=f(−2)=16 and 2≠−2
∴f is not one- one
For 2∈R there is no x in R such that f(x)=x4=2
∴f is not onto.
Hence, f is neither one-one nor onto.
Hence, option D is correct.
The number of all onto functions from the set {1,2,3,...,n} to itself is the permutations of n symbols 1,2,3,4,5...............n.
Hence, permutations on n symbols 1,2,3,4,5...............n = n
Thus, the total number of all onto maps from the set {1,2,3,...,n} to itself is the same as the permutations on n symbols 1,2,3,4,5...............n, which is n.
Question: Let an operation ∗ on the set of natural numbers N be defined by a∗b=ab⋅
Find (i) whether ∗ is a binary or not, and (ii) if it is a binary, then is it commutative or not.
Solution:
(i) As ab∈N for all a,b∈N
ie a∗b∈N∀a,b∈N
Hence, ∗ is binary.
Class 12 Maths NCERT Chapter 1, Relations and Functions: Important Formulae
Relations:
A relation R is a subset of the Cartesian product of A×B, where A and B are non-empty sets.
R−1, the inverse of relation R, is defined as:
R−1={(b,a):(a,b)∈R}
Domain of R= Range of R−1
Range of R= Domain of R−1
Functions:
A relation f from set A to set B is a function if every element in A has one and only one image in B.
A×B={(a,b):a∈A,b∈B}
If (a,b)=(x,y), then a=x and b=y
n(A×B)=n(A) * n(B), where n(A) is the cardinality (number of elements) of set A.
A×ϕ=ϕ (where ϕ is the empty set)
A function f:A→B is denoted as:
f(x)=y
This means (x,y)∈f.
Algebra of functions:
(f+g)(x)=f(x)+g(x)
(f−g)(x)=f(x)−g(x)
(f⋅g)(x)=f(x)⋅g(x)
(k⋅f)(x)=k⋅f(x), where k∈R
(f/g)(x)=f(x)/g(x), where g(x)≠0
Approach to Solve Questions of Relations and Functions Class 12
Relations and Functions can feel like a puzzle of definitions, diagrams, and properties—but with the right approach, every question starts to make perfect sense.
Recognising the type: Build a strong foundation for different kinds of relations, such as reflexive, symmetric, transitive, and equivalence.
Familiarise yourself with the properties of One-One, onto, and bijective functions.
Efficient usage of diagrams and graphs: Visual representation of the properties of relations and functions helps to clarify the types.
Use tables for small sets for binary operations to verify identity, inverse, etc.
Domain, codomain, range: Grasp these concepts thoroughly to identify whether a relation is a function and determine its behaviour.
Also, practice extracting the domain and range from relations using ordered pairs or graphs.
Elimination method: For MCQ-type questions, cancel out the extreme options by checking the basic definitions.
Shortcut tricks: Smart tricks and shortcuts are always handy during the exam to save time. Write those tricks in a notebook and revise them from time to time. Here are some tricks.
- If a relation is reflexive and only uses equality or divisibility, it may be an equivalence.
- For transitive relations, check in pairs that share a common middle.
- Use the formula 2n2 to find the total relations over a set of n elements.
- During the domain checking, avoid the values that cause division by zero or negative square roots.
What Extra Should Students Study Beyond the NCERT for JEE?
Q: How do you find the composition of functions in Chapter 1?
A:
In NCERT Class 12 Maths Chapter 1, the composition of functions means applying one function to the result of another. If we have two functions f(x) and g(x), their composition is written as (f o g)(x), which means f(g(x)).
Steps to find composition:
1. First, find g(x) (solve for x in g ).
2. Substitute g(x) into f(x).
3. Simplify the expression.
For example, if f(x)=x2 and g(x)=x+1, then:
(f o g)(x)=f(g(x))=f(x+1)=(x+1)2
Q: What is the range and domain of functions in NCERT Chapter 1?
A:
In NCERT Class 12 Maths Chapter 1, the domain and range of a function describe its input and output values:
Domain: The set of all possible input values (x) for which the function is defined. Example: For f(x)=1/x, the domain is all real numbers except x=0.
Range: The set of all possible output values f(x). Example: For f(x)=x2 , the range is [0, infinity) because squares are always non-negative.
Q: What is the difference between one-one and onto functions in NCERT Class 12 Maths?
A:
In NCERT Class 12 Maths, one-one (injective) and onto (surjective) functions are two important types of functions:
One-One Function (Injective): A function is one-one if different inputs give different outputs. This means no two elements in the domain map to the same element in the codomain. Example: f(x)=2x is one-one.
Onto Function (Surjective): A function is onto if every element in the codomain has at least one pre-image in the domain. This means the function covers the entire codomain. Example: f(x)=x3 is onto for real numbers.
A function can be both one-one and onto (bijective).
Q: What are the key topics covered in NCERT Class 12 Maths Chapter 1?
A:
NCERT Class 12 Maths Chapter 1 "Relations and Functions" covers essential standards of set principles and mappings. Key subjects encompass:
Types of Relations – Reflexive, symmetric, transitive, and equivalent members of the family.
Types of Functions – One-one (injective), onto (surjective), and bijective functions.
Composition of Functions – Understanding how functions integrate.
The inverse of a Function – Conditions for the lifestyles of an inverse.
Binary Operations – Definition consisting of commutativity and associativity.
Q: What is the difference between relation and function?
A:
In NCERT Class 12 Maths Chapter 1, relations and functions are different concepts:
Relation: A relation connects elements of one set to another. It is simply a pairing of elements, but it may not follow specific rules. Example: If Set A={1,2,3} and Set B={4,5}, a relation can be {(1,4), (2,5)}.
Function: A function is a special type of relation where each input has exactly one output. Example: f(x)=x+2 maps every x to a unique value.
Thus, every function is a relation, but not every relation is a function.
If you want to improve the Class 12 PCM results, you can appear in the improvement exam. This exam will help you to retake one or more subjects to achieve a better score. You should check the official website for details and the deadline of this exam.
SASTRA University commonly provides concessions and scholarships based on merit in class 12 board exams and JEE Main purposes with regard to board merit you need above 95% in PCM (or on aggregate) to get bigger concessions, usually if you scored 90% and above you may get partial concessions. I suppose the exact cut offs may change yearly on application rates too.
CBSE generally forwards the marksheet for the supplementary exam to the correspondence address as identified in the supplementary exam application form. It is not sent to the address indicated in the main exam form. Addresses that differ will use the supplementary exam address.
Yes, the Gujarat board does allow students to register as private candidates for class 12th examinations that means that even if you are not enrolled in any school you can sit for the examination. For this you will have to fill out a special private candidate application form and submit it along with relevant documents like marksheet, ID card, photos, etc to the Gujarat state education board office. You can submit it either offline or online.
However, the 2026 application windows has not opened yet and when it will typically schools will handle the registration through their board login so as of now you will have to reach out any GSEB affiliated school or board office itself in order to seek guidance on how to apply as a private candidate for 2026.
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