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Relations and Functions class 12 solutions are provided here. These NCERT solutions are created by expert team at careers360 keeping in mind of latest syllabus of CBSE 2023-24. This is the first chapter of Class 12 math. NCERT solutions class 12 maths chapter 1 Relations and Functions contains the answer and step-by-step solution to each question asked in the exercise of NCERT Class 12 maths book. NCERT Class 12 maths solutions Chapter 1 will help you to understand the concepts and score well in CBSE 12th board exam. Here you will find all NCERT solutions of chapter 1 maths class 12 at a single place which will be helpful when you are not able to solve the NCERT questions.
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In Relations and Functions class 12 maths chapter 1 question answer, there are four exercises with 55 questions and one miscellaneous exercise with 19 questions. relations and functions class 12 solutions are very important for students because they comprise quality practice questions. In this article, you will find the detailed NCERT solutions for class 12 maths chapter 1. Here you will get NCERT solutions for class 12 also.
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>> 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 ϵ A, b ϵ 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 of set A.
A × ϕ = ϕ (where ϕ is the empty set)
A function f: A → B is denoted as f(x) = y.
Algebra of functions:
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f * g)(x) = f(x) * g(x)
(kf)(x) = k * f(x), where k is a real number
{f/g}(x) = f(x)/g(x), where g(x) ≠ 0
Free download NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions for CBSE Exam.
NCERT Solutions for Class 12 relations and functions NCERT solutions: Exercise 1.1
Question1(i) . Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation
Answer:
Since,
Since,
Since,
Hence,
Question 1(ii) . Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set N of natural numbers defined as
Answer:
Since,
so
Since,
so
Since there is no pair in
Hence,
nor transitive.
Question1(iii) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iii) Relation R in the set
Answer:
Any number is divisible by itself and
Hence, it is transitive.
Hence, it is reflexive and transitive but not symmetric.
Question.1(iv) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iv). Relation R in the set Z of all integers defined as
Answer:
For
So,it is reflexive.
For
So, it is symmetric.
For
Now,
So,
Hence, it is reflexive, symmetric and transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
Answer:
Hence, it is reflexive, symmetric and transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
Answer:
It is same as
So,it is symmetric.
It implies that
Hence, it is reflexive, symmetric and
transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
Answer:
Hence, it is not reflexive,not symmetric and
not transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v). Relation R in the set A of human beings in a town at a particular time given by
Answer:
So, it is not reflexive.
So, it is not symmetric.
Let,
This case is not possible so it is not transitive.
Hence, it is not reflexive, symmetric and
transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
Answer:
Let,
So, it is not transitive.
Hence, it is neither reflexive nor symmetric and nor transitive.
Answer:
Taking
and
So, R is not reflexive.
Now,
But,
So,
Hence, it is not symmetric.
Since
Hence, it is not transitive.
Thus, we can conclude that it is neither reflexive, nor symmetric, nor transitive.
Question:3 Check whether the relation R defined in the set
Answer:
R defined in the set
Since,
So, it is not symmetric
So, it is not transitive.
Hence, it is neither reflexive, nor symmetric, nor transitive.
Question:4 Show that the relation R in R defined as
Answer:
As
Now we take an example
But
So,it is not symmetric.
Now if we take,
Than,
So, it is transitive.
Hence, we can say that it is reflexive and transitive but not symmetric.
Question:5 Check whether the relation R in R defined by
symmetric or transitive.
Answer:
So, it is not symmetric
Now,
but
It is not symmetric
But,
So it is not transitive
Thus, it is neither reflexive, nor symmetric, nor transitive.
Question:6 Show that the relation R in the set
symmetric but neither reflexive nor transitive.
Answer:
Let A=
We can see
As
But
Hence, R is symmetric but neither reflexive nor transitive.
Answer:
A = all the books in a library of a college
Let
Since y and x have the same number of pages so
Hence, it is symmetric.
Let
and
This states,x and z also have the same number of pages i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence
relation.?
Answer:
Let there be
Let
Hence, it is symmetric
Now, let
then,
So,
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The elements of
The elements of
The element of
Question:9(i) Show that each of the relation R in the set
(i)
Answer:
For
Henec, it is reflexive.
Let,
then
Hence, it is symmetric.
Let,
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is
Question:9(ii) Show that each of the relation R in the set
(ii)
Answer:
For
Henec, it is reflexive.
Let,
Hence, it is symmetric.
Let,
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is {1}
Question:10(i) Give an example of a relation.
(i) Which is Symmetric but neither reflexive nor transitive.
Answer:
Let
Hence, symmetric but neither reflexive nor transitive.
Question:10(ii) Give an example of a relation.
(ii) Which is transitive but neither reflexive nor symmetric.
Answer:
Let
Now for
Let
Then
Let
we can write this as
Hence,
Hence, it is transitive but neither reflexive nor symmetric.
Question:10(iii) Give an example of a relation.
(iii) Which is Reflexive and symmetric but not transitive.
Answer:
Let
Define a relation R on A as
If
If
But
Hence, it is Reflexive and symmetric but not transitive.
Question:10(iv) Give an example of a relation.
(iv) Which is Reflexive and transitive but not symmetric.
Answer:
Let there be a relation R in R
Let
But
So it is not symmetric.
Let
This can be written as
Hence, it is transitive.
Thus, it is Reflexive and transitive but not symmetric.
Question:10(v) Give an example of a relation.
(v) Which is Symmetric and transitive but not reflexive.
Answer:
Let there be a relation A in R
We can see
So it is symmetric.
Let
Also
Hence, it is transitive.
Thus, it Symmetric and transitive but not reflexive.
Answer:
The distance of point P from the origin is always the same as the distance of same point P from origin i.e.
Let
this is the same as distance of the point Q from the origin is the same as the distance of the point P from the origin i.e.
Let
i.e. the distance of point P from the origin is the same as the distance of point Q from the origin, and also the distance of 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 point P, Q, S from the origin is the same. Means distance of point P from the origin is the same as the distance of point S from origin i.e.
Hence, R is an equivalence relation.
The set of all points related to a point
In other words, we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from the origin.
Hence, these sets of points form a circle with the centre as the origin and this circle passes through the point.
Answer:
All triangles are similar to itself, so it is reflexive.
Let,
T 1 is similar to T2 is the same asT2 is similar to T 1 i.e.
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
Now, we see the ratio of sides of triangle T 1 andT 3 are as shown
i.e. ratios of sides of T 1 and T 3 are equal.Hence, T 1 and T 3 are related.
Answer:
The same polygon has the same number of sides with itself,i.e.
Let,
P 1 have the same number of sides as P 2 is the same as P 2 have same number of sides as P 1 i.e.
Hence,it is symmetric.
Let,
Hence, it is transitive,
Thus,
The elements in A related to the right angle triangle T with sides 3, 4 and 5 are those polygons which have 3 sides.
Hence, the set of all elements in A related to the right angle triangle T is set of all triangles.
Answer:
All lines are parallel to itself, so it is reflexive.
Let,
L1 is parallel to L 2 is same as L 2 is parallel to L 1 i.e.
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
The set of all lines related to the line
Here, Slope = m = 2 and constant = c = 4
It is known that the slope of parallel lines are equal.
Lines parallel to this (
Hence, set of all parallel lines to
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.
A = {1,2,3,4}
For every
Given,
For
Hence, R is reflexive and transitive but not symmetric.
The correct answer is option B.
(A) Since,
(B) Since,
(C) Since,
(d) Since,
The correct answer is option C.
Relations and Functions Class 12 NCERT Solutions: Exercise 1.2
Answer:
Given,
One - One :
Onto:
We have
Hence, the function is one-one and onto.
If the domain R ∗ is replaced by N with co-domain being same as R ∗ i.e.
For
Hence, function g is one-one but not onto.
Question:2(i) Check the injectivity and surjectivity of the following functions:
Answer:
One- one:
For
Hence, f is injective but not surjective.
Question:2(ii) Check the injectivity and surjectivity of the following functions:
Answer:
One- one:
For
For
Hence, f is neither injective nor surjective.
Question:2(iii) Check the injectivity and surjectivity of the following functions:
Answer:
One- one:
For
For
Hence, f is not injective and not surjective.
Question:2(iv) Check the injectivity and surjectivity of the following functions:
Answer:
One- one:
For
Hence, f is injective but not surjective.
Question:2(v) Check the injectivity and surjectivity of the following functions:
Answer:
One- one:
For
For
Hence, f is injective but not surjective.
Answer:
One- one:
For
but
For
Hence, f is not injective but not surjective.
Answer:
One- one:
For
For
We know
Hence,
Question:5 Show that the Signum Function
Answer:
As we can see
So it is not one-one.
Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain
So it is not onto.
Hence, signum function is neither one-one nor onto.
Question:6 Let
Answer:
Every element of A has a distant value in f.
Hence, it is one-one.
Question:7(i) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
Answer:
Let there be
Let there be
Puting value of x,
f is both one-one and onto hence, f is bijective.
Question:7(ii) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
Answer:
Let there be
For
Let there be
There does not exists any x in domain R such that
Hence, f is neither one-one nor onto.
Question:8 Let A and B be sets. Show that
bijective function.
Answer:
Let
such that
Let,
then there exists
Hence, it is bijective.
Question:9 Let
Answer:
Here we can observe,
As we can see
Let,
case1 n be even
For
then there is
case2 n be odd
For
then there is
f is not one-one but onto
hence, the function f is not bijective.
Question:10 Let
Answer:
Let
Let,
For any
Hence, the function is one-one and onto.
Question:11 Let
(D) f is neither one-one nor onto.
Answer:
One- one:
For
example: and
For
Hence, f is neither one-one nor onto.
Option D is correct.
Question:12 Let
(D) f is neither one-one nor onto.
Answer:
One - One :
Let
Onto:
We have
Hence, the function is one-one and onto.
The correct answer is A .
Relation and Function Class 12 maths chapter 1 question answer: Exercise 1.3
Question:1 Let
Answer:
Given :
Hence,
Question:4 If
Answer:
Hence,the given function
Question:5(i) State with reason whether following functions have inverse
(i)
with
Answer:
(i)
From the given definition,we have:
Hence, f do not have an inverse function.
Question:5(ii) State with reason whether following functions have inverse
Answer:
(ii)
From the definition, we can conclude :
Hence, function g does not have inverse function.
Question:5(iii) State with reason whether following functions have inverse
Answer:
(iii)
From the definition, we can see the set
For every element y of set
Thus, h is one-one and onto so h has an inverse function.
Question:6 Show that
Answer:
One -one:
It is clear that
Thus,f is one-one and onto so inverse of f exists.
Let g be inverse function of f in
let y be an arbitrary element of range f
Since,
Question:7 Consider
Answer:
One-one :
Let
Onto:
So, for
Thus, f is one-one and onto so
Let,
Now,
Hence, function f is invertible and inverse of f is
Answer:
It is given that
Now, Let f(x) = f(y)
⇒ x 2 + 4 = y 2 + 4
⇒ x 2 = y 2
⇒ x = y
⇒ f is one-one function.
Now, for y
⇒ x 2 = y -4 ≥ 0
⇒ for any y
= y -4 + 4 = y.
⇒ f is onto function.
Therefore, f is one–one and onto function, so f-1 exists.
Now, let us define g: [4, ∞) → R+ by,
g(y) =
Now, gof(x) = g(f(x)) = g(x 2 + 4) =
And, fog(y) = f(g(y)) = =
Therefore, gof = gof = I R .
Therefore, f is invertible and the inverse of f is given by
f-1(y) = g(y) =
Question:9 Consider
Answer:
One- one:
Let
Since, x and y are positive.
Onto:
Let for
Since f is one-one and onto so it is invertible.
Let
Hence,
Answer:
Let
Also, suppose f has two inverse
For
Thus,f has a unique inverse.
Question:11 Consider
Answer:
It is given that
Now,, lets define a function g :
Now,
Similarly,
And
Hence,
Therefore, the inverse of f exists and
Now,
Now, we need to find the inverse of
Therefore, lets define
Now,
Similarly,
Hence,
Therefore, inverse of
Therefore,
Hence proved
Question:12 Let
Answer:
To prove:
Let
Then there is
Also,
Hence,
i.e.
Question:14 Let
Answer:
Let f inverse
Let y be the element of range f.
Then there is
Now , define
Hence, g is inverse of f and
The inverse of f is given by
The correct option is B.
Class 12 maths chapter 1 NCERT solutions: Exercise 1.4
Answer:
(i) On
It is not a binary operation as the image of
Answer:
(ii) On
We can observe that for
This means * carries each pair
Therefore,* is a binary operation.
Answer:
(iii) On
We can observe that for
This means * carries each pair
Therefore,* is a binary operation.
Answer:
(iv) On
We can observe that for
This means * carries each pair
Therefore,* is a binary operation.
Answer:
(v) On
* carries each pair
Therefore,* is a binary operation.
Question:2(i) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
a*b=a-b
b*a=b-a
so * is not commutative
(a*b)*c=(a-b)-c
a*(b*c)=a-(b-c)=a-b+c
(a*b)*c not equal to a*(b*c), so * is not associative
Question:2(ii) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
(ii) On
ab = ba for all
ab+1 = ba + 1 for all
Question:2(iii) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
(iii) On
ab = ba for all
Question:2(iv) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
(iv) On
ab = ba for all
2ab = 2ba for all
Question:2(v) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
(v) On
Question:2(vi) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
Answer:
(iv) On
Question:3
Consider the binary operation Λ on the set {1, 2, 3, 4, 5} defined by
a Λ b = min {a, b}. Write the operation table of the operation Λ .
Answer:
The operation table of the operation
| 1 | 2 | 3 | 4 | 5 |
1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 2 | 2 | 2 |
3 | 1 | 2 | 3 | 3 | 3 |
4 | 1 | 2 | 3 | 4 | 4 |
5 | 1 | 2 | 3 | 4 | 5 |
Question:4(i) Consider a binary operation ∗ on the set
(Hint: use the following table)
Answer:
(i)
Question:4(ii) Consider a binary operation ∗ on the set
(Hint: use the following table)
Answer:
(ii)
For every
Question:4(iii) Consider a binary operation ∗ on the set {
(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).
(Hint: use the following table)
Answer:
(iii) (2 ∗ 3) ∗ (4 ∗ 5).
from the above table
Answer:
The operation table is as shown below:
| 1 | 2 | 3 | 4 | 5 |
1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 1 | 2 | 1 |
3 | 1 | 1 | 3 | 1 | 1 |
4 | 1 | 2 | 1 | 4 | 1 |
5 | 1 | 1 | 1 | 1 | 5 |
The operation ∗′ same as the operation ∗ defined in Exercise 4 above.
Question:6(i) let ∗ be the binary operation on N given by . Find
Answer:
a*b=LCM of a and b
(i) 5 ∗ 7, 20 ∗ 16
Question:6(ii) Let ∗ be the binary operation on N given by
Answer:
(ii)
Hence, it is commutative.
Question:6(iii) Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find
Answer:
a
(iii)
Hence, the operation is associative.
Question:6(iv) Let ∗ be the binary operation on N given by
Answer:
(iv) the identity of ∗ in N
We know that
Hence, 1 is the identity of ∗ in N.
Question 6(v) Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find
(v) Which elements of N are invertible for the operation ∗?
Answer:
An element a is invertible in N
if
Here a is inverse of b.
a*b=1=b*a
a*b=L.C.M. od a and b
a=b=1
So 1 is the only invertible element of N
Question:7 Is ∗ defined on the set
Answer:
A =
Operation table is as shown below:
| 1 | 2 | 3 | 4 | 5 |
1 | 1 | 2 | 3 | 4 | 5 |
2 | 2 | 2 | 6 | 4 | 10 |
3 | 3 | 6 | 3 | 12 | 15 |
4 | 4 | 4 | 12 | 4 | 20 |
5 | 5 | 10 | 15 | 20 | 5 |
From the table, we can observe that
Hence, the operation is not a binary operation.
Answer:
a ∗ b = H.C.F. of a and b for all
H.C.F. of a and b = H.C.F of b and a for all
Hence, operation ∗ is commutative.
For
Hence, ∗ is associative.
An element
Hence, the operation * does not have any identity in N.
Question:9(i) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(i)
Answer:
On the set Q ,the operation * is defined as
Hence, the * operation is not commutative.
It can be observed that
The operation * is not associative.
Question:9(ii) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(ii)
Answer:
On the set Q ,the operation * is defines as
For
Hence, the * operation is commutative.
It can be observed that
The operation * is not associative.
Question:9(iii) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(iii)
Answer:
On the set Q ,the operation * is defines as
For
Hence, the * operation is not commutative.
It can be observed that
The operation * is not associative.
Question:9(iv) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(iv)
Answer:
On the set Q ,the operation * is defined as
For
Hence, the * operation is commutative.
It can be observed that
The operation * is not associative.
Question:9(v) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(v)
Answer:
On the set Q ,the operation * is defines as
For
Hence, the * operation is commutative.
It can be observed that
The operation * is associative.
Question:9(vi) Let ∗ be a binary operation on the set Q of rational numbers as follows:
(vi)
Answer:
On the set Q ,the operation * is defines as
For
Hence, the * operation is not commutative.
It can be observed that
The operation * is not associative.
Question:10 Find which of the operations given above has identity.
Answer:
An element
if
Hence,
However, there is no such element
Hence, only (v) operations have identity.
Question:11 Let
Answer:
Let
Then,
We have
Thus it is commutative.
Let
Then,
Thus, it is associative.
Let
i.e.
This is not possible for any element in A .
Hence, it does not have any identity.
Question:12(i) State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation ∗ on a set N,
Answer:
(i) For an arbitrary binary operation ∗ on a set N,
An operation * on a set N as
Then , for b=a=2
Hence, statement (i) is false.
Question:12(ii) State whether the following statements are true or false. Justify.
(ii) If ∗ is a commutative binary operation on N, then
Answer:
(ii) If ∗ is a commutative binary operation on N, then
R.H.S
= L.H.S
Hence, statement (ii) is true.
Question:13 Consider a binary operation ∗ on N defined as
Answer:
A binary operation ∗ on N defined as
For
Thus, it is commutative.
Hence, it is not associative.
Hence, B is the correct option.
NCERT solutions for class 12 maths chapter 1 Relations and Functions: Miscellaneous Exercise
Question:1 Let
Answer:
For one-one:
Thus, f is one-one.
For onto:
For
Thus, for
Thus, f is onto.
Hence, f is one-one and onto i.e. it is invertible.
Let
Hence,
Answer:
For one-one:
Taking x as odd number and y as even number.
Now, Taking y as odd number and x as even number.
This is also impossible.
If both x and y are odd :
If both x and y are even :
Onto:
Any odd number 2r+1 in codomain of N is an image of 2r in domain N and any even number 2r in codomain N is the image of 2r+1 in domain N.
Thus, f is onto.
Hence, f is one-one and onto i.e. it is invertible.
Sice, f is invertible.
Let
When x is odd.
When x is even
Similarly, m is odd
m is even ,
Hence, f is invertible and the inverse of f is g i.e.
Hence, inverse of f is f itself.
Question:3 If f : R → R is defined by f(x) = x 2 – 3x + 2, find f (f (x)).
Answer:
This can be solved as following
f : R → R
Question:4 Show that the function
Answer:
The function
One- one:
Let
It is observed that if x is positive and y is negative.
Since x is positive and y is negative.
Thus, the case of x is positive and y is negative is removed.
Same happens in the case of y is positive and x is negative so this case is also removed.
When x and y both are positive:
When x and y both are negative :
Onto:
Let
If y is negative, then
If y is positive, then
Thus, f is onto.
Hence, f is one-one and onto.
Question:5 Show that the function
Answer:
One-one:
Let
We need to prove
Let
It will contradict given condition of cubes being equal.
Hence,
Question:6 Give examples of two functions
Answer:
One - one:
Since
As we can see
Thus , g(x) is not injective.
Let
Since,
Hence, gof is injective.
Question:7 Give examples of two functions
Answer:
Onto :
Consider element in codomain N . It is clear that this element is not an image of any of element in domain N .
Now, it is clear that
Hence,
Answer:
Given a non empty set X, consider P(X) which is the set of all subsets of X.
Since, every set is subset of itself , ARA for all
Let
This is not same as
If
then we cannot say that B is related to A.
If
this implies
Thus, R is not an equivalence relation because it is not symmetric.
Answer:
Given
As we know that
Hence, X is the identity element of binary operation *.
Now, an element
such that
i.e.
This is possible only if
Hence, X is only invertible element in
Question:10 Find the number of all onto functions from the set
Answer:
The number of all onto functions from the set
Hence, permutations on n symbols 1,2,3,4,5...............n = n
Thus, total number of all onto maps from the set
Question:11(i) Let
Answer:
Question:11(ii) Let
Answer:
So inverse of F does not exists.
Hence, F is not invertible i.e.
Answer:
Given
For
Let
Hence,
Now,
Hence, operation o does not distribute over operation *.
Question:13 Given a non-empty set X, let ∗ : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation ∗ and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).
Answer:
It is given that
Now, let
Then,
And
Therefore,
Therefore, we can say that
Now, an element A
Now, We can see that
Therefore, by this we can say that all the element A of P(X) are invertible with
Question:14 Define a binary operation ∗ on the set
Answer:
X =
An element
For
Hence, 0 is identity element of operation *.
An element
such that
means
But since we have X =
Hence, inverse of element
, is 6-a i.e. ,
Question:15 Let
Answer:
Given :
It can be observed that
Hence, f and g are equal functions.
Question:16 Let
Answer:
The smallest relations containing
reflexive and symmetric but not transitive is given by
Now, if we add any two pairs
Hence, the total number of the desired relation is one.
Thus, option A is correct.
Question:17 Let
(A) 1
(B) 2
(C) 3
(D) 4
Answer:
The number of equivalence relations containing
We are left with four pairs
Hence , equivalence relation is bigger than R is universal relation.
Thus the total number of equivalence relations cotaining
Thus, option B is correct.
Question:18 Let
Answer:
Let
Then we have ,
Hence,for
Hence , gof and fog do not coincide with
Question:19 Number of binary operations on the set are(A) 10(B) 16(C) 20(D ) 8
Answer:
Binary operations on the set
i.e. * is a function from
Hence, the total number of binary operations on set
Hence, option B is correct
If you are looking for relation and function class 12 ncert solutions of exercises then these are listed below.
Relation And Function Class 12 Ncert Solutions Exercise 1.1
Relation And Function Class 12 Ncert Solutions Exercise 1.2
Relation And Function Class 12 Ncert Solutions Exercise 1.3
Relation And Function Class 12 Ncert Solutions Exercise 1.4
Relation And Function Class 12 Ncert Solutions Miscellaneous Exercise
We have relations like father, mother, brother, sister, husband, wife. Relation becomes a function when there is only one output for every input. In NCERT class 11 maths solutions you have already learnt in brief about relations and functions, range, domain and co-domain with different types of specific real-valued functions and their graphs.
In Class 12 maths chapter 1 question answer, you will learn about different types of relations and functions, invertible functions, the composition of functions, and binary operations. Also you can find ncert solutions for class 12 chapter 1 by careers360 expert team.
Concepts of this chapter are very useful in various other topics of calculus and are also very important from the exam point of view. Unit "Relation and Function" of NCERT class 12th maths syllabus includes two chapters i.e. relation and function, and inverse trigonometry which together has 10 % weightage in the CBSE class 12th final examination. So, you should study class 12 maths ch 1 question answer carefully, and solve every question on your own including solved examples.
NCERT Exemplar Solutions Chapter 1 - Relations and Functions
NCERT Exemplar Solutions Chapter 2 - Inverse Trigonometric Functions
NCERT Exemplar Solutions Chapter 3 - Matrices
NCERT Exemplar Solutions Chapter 4 - Determinants
NCERT Exemplar Solutions Chapter 5 - Continuity and Differentiability
NCERT Exemplar Solutions Chapter 6 - Application of Derivatives
NCERT Exemplar Solutions Chapter 7 - Integrals
NCERT Exemplar Solutions Chapter 8 - Application of Integrals
NCERT Exemplar Solutions Chapter 9 - Differential Equations
NCERT Exemplar Solutions Chapter 10 - Vector Algebra
NCERT Exemplar Solutions Chapter 11 - Three Dimensional Geometry
NCERT Exemplar Solutions Chapter 12 - Linear Programming
NCERT Exemplar Solutions Chapter 13 - Probability
These Relations and Functions class 12 NCERT solutions are explained in a step-by-step method, so it will be very easy to understand the concepts. Still if you are in a doubt anywhere, you can contact our subject matter experts who are available to help you out and make learning easier for you.
The meaning of the term ‘relation’ in mathematics is the same as the meaning of ' relation' in the English language. Relation means two quantities or objects are related if there is a link between them. In other words, we can say that it is a connection between or among things.
Let's understand with an example - let A is the set of students of class XII of a school and B is the set of students of class XI of the same school. Then some of the examples of relations from A to B are-
(i) {(a, b) ∈A × B: a is a brother of b},
(ii) {(a, b) ∈A × B: a is a sister of b},
(iii) {(a, b) ∈A × B: age of a is less than the age of b}.
If (a, b) ∈ R, we can say that ‘a’ is related to ‘b’ under the relation ‘R’ and we write as ‘a R b’. To understand the topic in-depth, after every concept, some topic wise questions are given in the textbook of CBSE class 12. In this article, you will find solutions of NCERT for class 12 maths chapter 1 Relations and Functions for such type of questions also.
The main topics covered in chapter 1 maths class 12 are:
In this ch 1 maths class 12 topics discuss different types of relations namely reflexive, symmetric, and transitive. we also study the concept of empty relation, universal relation, trivial relation, and equivalence relation in the chapter relations and functions. there are good quality questions in functions and relations class 12 solutions.
This ch 1 maths class 12 concerns different types of functions like constant function, polynomial function, identity function, rational function, modulus function, signum function, etc. this chapter also contains the concept of one-one (or injective), onto (or subjective), one-one and onto (or bijective) functions. The concept of addition, subtraction, multiplication, and division of two functions have also been discussed. to get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 1.
we understand the concept of composition of a function in this chapter of class 12 NCERT. also we get a good hold of invertible functions concepts in this chapter. for questions, you can browse class 12 NCERT solutions.
this ch 1 maths class 12 also includes concepts of binary operations. terms like commutative, associative invertible, inverse, identity are also discussed in class 12 NCERT. you can refer to class 12 NCERT solutions for questions about these concepts.
Topics enumerated in class 12 NCERT are very important and students are advised to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the class 12 maths ch 1 question answer.
Also read,
Chapter 1 | Relations And Functions |
Chapter 2 | |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 |
Class 12 relations and functions NCERT solutions is helpful for the students who wish to perform well in the CBSE 12 board examination. Some guidelines to follow to make the best use of NCERT solutions:
Happy learning !!!
As CBSE board exam paper is designed entirelly based on NCERT textbooks and most of the questions in CBSE board exam are directly asked from NCERT textbook, students must know the relations and functions class 12 questions and answers very well to perform well in the exam. NCERT solutions are not only important when you stuck while solving the problems but students will get how to answer in the board exam in order to get good marks in the board exam.
Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this maths chapter 1 class 12. these topics are important because concepts are used in calculus and other topics as well as exams therefore students are recommended ncert solutions and ncert exercise to get command on the concepts.
The NCERT Solutions for maths chapter 1 class 12 provide in-depth explanations of several important concepts, including types of relations, different types of functions, composition of functions, invertible functions, and binary operations. These solutions for class 12 maths ch 1 are created by a team of highly qualified and experienced teachers and their primary goal is to assist students in achieving a high score on their Class 12 Maths board exams.
Following are some key attributes of relation and function class 12 solutions.
The maths chapter 1 class 12 are created by experienced subject matter experts who possess a deep understanding of the key concepts.
The solutions for relation function class 12 are presented in a clear and straightforward language to make it easy for students to grasp the methods for solving complex problems.
The step-by-step solutions for class 12 relation and function are designed based on the marks weightage assigned by the CBSE exam, ensuring that students can maximize their scores on the exam.
The Maths class 12 relations and functions ncert solutions are updated with the latest CBSE guidelines, ensuring that all the important topics are covered. The chapter contains four exercises, offering students a variety of problems to solve on their own. The class 12 maths ch 1 ncert solutions are structured to build confidence in students ahead of the CBSE exams. For ease, students can study relations and functions class 12 ncert pdf online and offline in both modes.
Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
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I hope this information helps you.
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If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
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