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Install NowNCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.
Download NCERT solutions for Relations and Functions as PDF.
NCERT Solutions class 12 Maths Relations and Functions
1. Let 
be defined as 
Find the function 
such that 
Ans. Given: 
Now 
and 
2. Let 
be defined as 
if 
is odd and 
if is even. Show that 
is invertible. Find the inverse of 
Here, W is the set of all whole numbers.
Ans. Given: defined as 
Injectivity: Let 
be any two odd real numbers, then 
Again, let be any two even whole numbers, then
Is is even and 
is odd, then 
Also, if 
odd and 
is even, then 
Hence,
is an injective mapping.
Surjectivity: Let be an arbitrary whole number.
If is an odd number, then there exists an even whole number 
such that
If is an even number, then there exists an odd whole number 
such that
Therefore, every 
W has its pre-image in W.
So, is a surjective. Thus is invertible and 
exists.
For : 
and 
Hence, 
NCERT Solutions class 12 Maths Miscellaneous
3. If is defined by 
find 
Ans. Given: 
= 
NCERT Solutions class 12 Maths Miscellaneous
4. Show that the function 
defined by 
R is one-one and onto function.
Ans. is one-one: For any 
R – {+1}, we have 
Therefore, is one-one function.
If is one-one, let 
R – {1}, then 
It is cleat that R for all R – {1}, also 
Because 
which is not possible.
Thus for each R – {1} there exists 
R – {1} such that
Therefore is onto function.
NCERT Solutions class 12 Maths Miscellaneous
5. Show that the function given by 
is injective.
Ans. Let 
R be such that 
Therefore, is one-one function, hence is injective.
NCERT Solutions class 12 Maths Miscellaneous
6. Give examples of two functions 
and 
such that 
is injective but 
is not injective.
(Hint: Consider 
and 
)
Ans. Given: two functions and
Let and 
Therefore, is injective but is not injective.
NCERT Solutions class 12 Maths Miscellaneous
7. Give examples of two functions 
and 
such that is onto but is not onto.
(Hint: Consider 
and 
)
Ans. Let
These are two examples in which is onto but is not onto.
NCERT Solutions class 12 Maths Miscellaneous
8. Given a non empty set X, consider P (X) which is the set of all subsets of X.
Define the relation AR in P (X) as follows:
For subsets A, B in P (X), ARB if and only if A
B. Is R an equivalence relation on P (X)? Justify your answer.
Ans. (i) A A R is reflexive.
(ii) A B 
B A R is not commutative.
(iii) If A B, B C, then A C R is transitive.
Therefore, R is not equivalent relation.
9. Given a non-empty set X, consider the binary operation * : P (X) x P (X) 
P (X) given by A * B = A 
B 
A, B in P (X), where P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation *.
Ans. Let S be a non-empty set and P(S) be its power set. Let any two subsets A and B of S.
A 
B S
A B 
P(S)
Therefore, 
is an binary operation on P(S).
Similarly, if A, B P(S) and A – B P(S), then the intersection of sets and difference of sets are also binary operation on P(S) and A S = A = S A for every subset A of sets
A S = A = S A for all A P(S)
S is the identity element for intersection 
on P(S).
10. Find the number of all onto functions from the set {1, 2, 3, ……., } to itself.
Ans. The number of onto functions that can be defined from a finite set A containing elements onto a finite set B containing elements = 
NCERT Solutions class 12 Maths Miscellaneous
11. Let S = 
and T = {1, 2, 3}. Find 
of the following functions F from S to T, if it exists.
(i) F = 
(ii) F = 
Ans. S = and T = {1, 2, 3}
(i) F =
(ii)
F is not one-one function, since element 
and 
have the same image 1.
Therefore, F is not one-one function.
NCERT Solutions class 12 Maths Miscellaneous
12. Consider the binary operation * : R x R R and o = R x R R defined as 
and 
R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that 
R, 
[If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Ans. Part I: also 
operation * is commutative.
Now, 
And 
Here, 
operation * is not associative.
Part II: 
R
And, 
operation 
is not commutative.
Now 
and 
Here 
operation is associative.
Part III: L.H.S. 
=
R.H.S. 
= 
= L.H.S. Proved.
Now, another distribution law: 
L.H.S. 
R.H.S. 
As L.H.S. R.H.S.
Therefore, the operation does not distribute over.
NCERT Solutions class 12 Maths Miscellaneous
13. Given a non-empty set X, let * : P (X) x P (X) P (X) be defined as A * B = (A – B) (B – A), A, B P (X). Show that the empty set 
is the identity for the operation * and all the elements A of P (X) are invertible with A-1 = A. (Hint: 
and 
)
Ans. For every A P(X), we have
= 
And 
= 
is the identity element for the operation * on P(X).
Also A * A = (A – A) (A – A) = 
Every element A of P(X) is invertible with 
= A.
NCERT Solutions class 12 Maths Miscellaneous
14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as 
Show that zero is the identity for this operation and each element 
of the set is invertible with 
being the inverse of 
Ans. A binary operation (or composition) * on a (non-empty) set is a function * : A x A A. We denote 
by 
for every ordered pair 
A x A.
A binary operation on a no-empty set A is a rule that associates with every ordered pair of elements 
(distinct or equal) of A some unique element of A.
* | 0 | 1 | 2 | 3 | 4 | 5 |
0 | 0 | 1 | 2 | 3 | 4 | 5 |
1 | 1 | 2 | 3 | 4 | 5 | 0 |
2 | 2 | 3 | 4 | 5 | 0 | 1 |
3 | 3 | 4 | 5 | 0 | 1 | 2 |
4 | 4 | 5 | 0 | 1 | 2 | 3 |
5 | 5 | 0 | 1 | 2 | 3 | 4 |
For all 
A, we have 
(mod 6) = 0
And 
and 
0 is the identity element for the operation.
Also on 0 = 0 – 0 = 0 *
2 * 1 = 3 = 1 * 2
NCERT Solutions class 12 Maths Miscellaneous
15. Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and 
be the functions defined by 
A and 
A. Are and equal? Justify your answer.
(Hint: One may note that two functions 
and 
such that 
A, are called equal functions).
Ans. When then 
and 
At 
and 
At 
and 
At 
and 
Thus for each A,
Therefore, and are equal function.
NCERT Solutions class 12 Maths Miscellaneous
16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:
(A) 1
(B) 2
(C) 3
(D) 4
Ans. It is clear that 1 is reflexive and symmetric but not transitive.
Therefore, option (A) is correct.
17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:
(A) 1
(B) 2
(C) 3
(D) 4
Ans. 2
Therefore, option (B) is correct.
NCERT Solutions class 12 Maths Miscellaneous
18. Let be the Signum Function defined as 
and 
be the Greatest Function given by 
where 
is greatest integer less than or equal to 
Then, does 
and coincide in (0, 1)?
Ans. It is clear that 
and 
Consider 
which lie on (0, # 1)
Now, 
And 
in (0, 1]
Therefore, option (B) is correct.
NCERT Solutions class 12 Maths Miscellaneous
19. Number of binary operation on the set 
are:
(A) 10
(b) 16
(C) 20
(D) 8
Ans. A =
A x A = 
= 4
Number of subsets = 
= 16
Hence number of binary operation is 16.
Therefore, option (B) is correct.
NCERT Solutions class 12 Maths Miscellaneous
NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide
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