NCERT Solutions for Class 12th: Ch 1 Relations and Functions Miscellaneous Exercise

NCERT Solutions for Class 12th: Ch 1 Relations and Functions Miscellaneous Exercise Math

Page No: 29

Exercise 1.3

1. Let f : R → R be defined as f(x) = 10x + 7. Find the function g : R → R such that gof = fog = IR .

Answer

It is given that f: R → R is defined as f(x) = 10x + 7.
For one – one
Let f(x) = f(y), where x, y ∈ R.
⇒ 10x + 7 = 10y + 7
⇒ x = y
∴ f is a one – one function.
For onto
For y ∈ R, let y = 10x + 7.
∴ f is onto.
Therefore, f is one–one and onto.
Thus, f is an invertible function.
Let us define g: R → R as g(y) = y−7/10
Now, we have

2. Let f : W → W be defined as f(n) = n–1, if n is odd and f(n) = n+1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

Answer

It is given that:
For one–one
Let f(n) = f(m).
It can be observed that if n is odd and m is even, then we will have n−1 = m + 1.
⇒ n − m = 2
However, this is impossible.
Similarly, the possibility of n being even and m being odd can also be ignored
under a similar argument.
∴ Both n and m must be either odd or even. Now, if both n and m are odd,
Then, we have
f(n) = f(m)
⇒ n − 1 = m – 1
⇒ n = m
Again, if both n and m are even,
Then, we have
f(n) = f(m)
⇒ n + 1 = m + 1
⇒ n = m
∴ f is one–one.

For onto:
It is clear that any odd number 2r + 1 in co-domain N is the image of 2r in domain
N and any even number 2r in co-domain N is the image of 2r + 1 in domain N.
∴ f is onto.
Hence, f is an invertible function.
Now, when n is odd
gof(n) = g(f(n)) = g(n − 1) = n − 1 + 1 = n and
When n is even
gof(n) = g(f(n)) = g(n + 1) = n + 1 − 1 = n
Similarly,
When m is odd
fog(m) = f(g(m)) = f(m − 1) = m − 1 + 1 = m and
When m is even
fog(m) = f(g(m)) = f(m + 1) = m + 1 − 1 = m
∴ gof = IW and fog = IW
Thus, f is invertible and the inverse of f is given by f-1 = g, which is the same as f. Hence, the inverse of f is f itself.

3. If f : R → R is defined by f(x) = x2 - 3x + 2, find f (f(x)).

Answer

It is given that f: R → R is defined as f(x) = x2 − 3x + 2.
f(f(x)) = f(x2 − 3x + 2)
= (x2 − 3x + 2)2 − 3(x2 − 3x + 2) + 2
= (x4 + 9x2 + 4 − 6x3 − 12x + 4x2) + (−3x2 + 9x − 6) + 2
= x4 − 6x3 + 10x2 − 3x

4. Show that the function f : R → {x ∈ R : –1 < x < 1} defined by
 , x ∈ R is one one and onto function.

Answer

For one – one
Suppose f(x) = f(y), where x, y ∈ R.
Since, x is positive and y is negative
x > y ⇒ x − y > 0
But, 2xy is negative.
Then 2xy ≠ x − y

Thus, the case of x being positive and y being negative can be ruled out.
Under a similar argument, x being negative and y being positive can also be ruled out.
∴ x and y have to be either positive or negative.
When x and y are both positive, we have
∴ f is one – one.
For onto
Now, let y ∈ R such that −1 < y < 1.
If y is negative, then, there exists x =y/1+y ∈ R such that
∴ f is onto.
Hence, f is one–one and onto.

5. Show that the function f : R → R given by f(x) = x3 is injective.

Answer

f: R → R is given as f(x) = x3.

For one – one
Suppose f(x) = f(y), where x, y ∈ R.
⇒ x3 = y3 ....(1)
Now, we need to show that x = y.
Suppose x ≠ y, their cubes will also not be equal.
⇒ x3 ≠ y3
However, this will be a contradiction to (1).
∴ x = y
Hence, f is injective.

6. Give examples of two functions f : N → Z and g : Z → Z such that g o f is injective but g is not injective.
(Hint : Consider f(x) = x and g (x) = | x |). 

Answer

Define f: N → Z as f(x) = x and g: Z → Z as g (x) = |x|.
We first show that g is not injective.
It can be observed that
g (−1) =|−1| = 1
g (1) = |1| = 1
∴ g (−1) = g (1), but −1 ≠ 1.
∴ g is not injective.
Now, g of: N → Z is defined as gof(x) = g(f(x)) = g(x) = |x|.
Let x, y ∈ N such that g of(x) = g of(y).
⇒ |x| = |y|
Since x and y ∈ N, both are positive.
∴ |x| = |y| ⇒ x = y
Hence, gof is injective.

7. Give examples of two functions f : N → N and g : N → N such that g o f is onto but f is not onto.

Answer

Define f: N → N by f(x) = x + 1 and,
We first show that g is not onto.
For this, consider element 1 in co-domain N. It is clear that this element is not an image of any of the elements in domain N.
∴ f is not onto.
Now, gof: N → N is defined by
gof(x) = g(f(x)) = g(x + 1) = x + 1 − 1 = x [x ∈ N ⇒ x + 1 > 1]
Then, it is clear that for y ∈ N, there exists x = y ∈ N such that gof(x) = y.
Hence, gof is onto.

8. Given a non empty set X, consider P(X) which is the set of all subsets of X.
Define the relation R 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.

Answer

Since every set is a subset of itself, ARA for all A ∈ P(X).
∴ R is reflexive.
Let ARB ⇒ A ⊂ B.
This cannot be implied to B ⊂ A.
For instance, if A = {1, 2} and B = {1, 2, 3}, then it cannot be implied that B is
related to A.
∴ R is not symmetric.
Further, if ARB and BRC, then A ⊂ B and B ⊂ C.
⇒ A ⊂ C
⇒ ARC
∴ R is transitive.
Hence, R is not an equivalence relation as it is not symmetric.

Page No. 30

9. Given a non-empty set X, consider the binary operation ∗ : P(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 ∗.

Answer

It is given the binary operation *:
P(X) × P(X) → P(X) given by A * B = A ∩ B ∀ A, B in P(X)
We know that A ∩ X = A = X ∩ A for all A ∈ P(X)
⇒ A * X = A = X * A for all A ∈ P(X)
Thus, X is the identity element for the given binary operation *.
Now, an element A ∈ P(X) is invertible if there exists B ∈ P(X) such that
A * B = X = B * A [As X is the identity element]
or
A ∩ B = X = B ∩ A
This case is possible only when A = X = B.
Thus, X is the only invertible element in P(X) with respect to the given operation*.
Hence, the given result is proved.

10. Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.

Answer

Onto functions from the set {1, 2, 3, ... , n} to itself is simply a permutation on n symbols 1, 2, ..., n.
Thus, the total number of onto maps from {1, 2, ... , n} to itself is the same as the total number of permutations on n symbols 1, 2, ..., n, which is n.

11. Let S = {a, b, c} and T = {1, 2, 3}. Find F-1 of the following functions F from S to T, if it exists.
(i) F = {(a, 3), (b, 2), (c, 1)}
(ii) F = {(a, 2), (b, 1), (c, 1)}

Answer

S = {a, b, c}, T = {1, 2, 3}
(i) F: S → T is defined as F = {(a, 3), (b, 2), (c, 1)}
⇒ F (a) = 3, F (b) = 2, F(c) = 1
Therefore, F-1 : T → S is given by F-1 = {(3, a), (2, b), (1, c)}.

(ii) F: S → T is defined as F = {(a, 2), (b, 1), (c, 1)}
Since F (b) = F (c) = 1, F is not one – one.
Hence, F is not invertible i.e., F-1 does not exist.

12. Consider the binary operations ∗ : R × R → R and o : R × R → R defined as a ∗b = |a – b| and a o b = a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a ∗ (b o c) = (a ∗ b) o (a ∗ c). [If it is so, we say that the operation ∗ distributes over the operation o]. Does o distribute over ∗? Justify your answer.

Answer

It is given that *: R × R → and o: R × R → R is defined as a ∗ b = |a − b| and
aob = a, ∀ a, b ∈ R
For a, b ∈ R, we have
a ∗ b = |a − b| and b ∗ a = |b − a| = |−(a − b)| = |a − b|
∴ a * b = b * a
Hence, the operation * is commutative.
It can be observed that
(1 ∗ 2) ∗ 3 = (|1 − 2|) ∗ 3 = 1 ∗ 3 = |1 − 3| = 2
and
1 ∗ (2 ∗ 3) = 1 ∗ (|2 − 3|) = 1 ∗ 1 = |1 − 1| = 0
∴ (1 ∗ 2) ∗ 3 ≠ 1 ∗ (2 ∗ 3) where 1, 2, 3 ∈ R.
Hence, the operation * is not associative.
Now, consider the operation o
It can be observed that 1 o 2 = 1 and 2 o 1 = 2.
∴ 1 o 2 ≠ 2 o 1 where 1, 2 ∈ R.
Hence, the operation o is not commutative.
Let a, b, c ∈ R. Then, we have
(a o b) o c = a o c = a
and
a o (b o c) = a o b = a
∴ (a o b) o c = a o (b o c), where a, b, c ∈ R
Hence, the operation o is associative.
Now, let a, b, c ∈ R, then we have
a * (b o c) = a * b =|a − b|
(a * b) o (a * c) = (|a − b|)o(|a − c|) = |a − b|
Hence, a * (b o c) = (a * b) o (a * c).
Now,
1 o (2 ∗ 3) = 1o(|2 − 3|) = 1o1 = 1
(1 o 2) * (1 o 3) = 1 * 1 = |1 − 1| = 0
∴ 1 o (2 * 3) ≠ (1 o 2) * (1 o 3) where 1, 2, 3 ∈ R
Hence, the operation o does not distribute over *.

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 *: P(X) × P(X) → P(X) is defined
as A * B = (A − B) ∪ (B − A) ∀ A, B ∈ P(X).
Let A ∈ P(X). Then, we have
A * Φ = (A − Φ) ∪ (Φ − A) = A ∪ Φ = A
Φ * A = (Φ − A) ∪ (A − Φ) = Φ ∪ A = A
∴ A * Φ = A = Φ * A for all A ∈ P(X)
Thus, Φ is the identity element for the given operation*.
Now, an element A ∈ P(X) will be invertible if there exists B ∈ P(X) such that
A * B = Φ = B * A. [As Φ is the identity element]
Now, we observed that,
A * A = ( A – A) ∪ (A – A) = Φ ∪ Φ = Φ for all A ∈ P(X).
Hence, all the elements A of P(X) are invertible with A-1 = A.

14. Define a binary operation ∗ on the set {0, 1, 2, 3, 4, 5} as
Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6–a being the inverse of a

Answer

Let X = {0, 1, 2, 3, 4, 5}.
An element e ∈ X is the identity element for the operation *, if
a ∗ e = a = e ∗ a for all a ∈ X
For a ∈ X, we have
a ∗ 0 = a + 0 = a [a ∈ X ⇒ a + 0 < 6]
0 ∗ a = 0 + a = a [a ∈ X ⇒ 0 + a < 6]
∴ a ∗ 0 = a = 0 ∗ a for all a ∈ X
Thus, 0 is the identity element for the given operation *.

An element a ∈ X is invertible if there exists b ∈ X such that a * b = 0 = b * a.
⇒ a = −b or b = 6 − a
But, X = {0, 1, 2, 3, 4, 5} and a, b ∈ X. Then, a ≠ −b.
∴ b = 6 − a is the inverse of a for all a ∈ X.
Hence, the inverse of an element a ∈X, a ≠ 0 is 6 − a i.e., a-1 = 6 − a.

15. Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined by
Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g (a) ∀ a ∈ A, are called equal functions).

Answer

It is given that A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2}.
Also, it is given that f, g: A → B are defined by
∴ f(a) = g(a) for all a ∈ A
Hence, the functions f and g are equal.

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

Answer

The given set is A = {1, 2, 3}.
The smallest relation containing (1, 2) and (1, 3) which is reflexive and symmetric, but not transitive is given by:
R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)}
This is because relation R is reflexive as (1, 1), (2, 2), (3, 3) ∈ R.
Relation R is symmetric since (1, 2), (2, 1) ∈ R and (1, 3), (3, 1) ∈ R.
But relation R is not transitive as (3, 1), (1, 2) ∈ R, but (3, 2) ∉ R.
Now, if we add any two pairs (3, 2) and (2, 3) (or both) to relation R, then relation R will become transitive.
Hence, the total number of desired relations is one.

The correct answer is A.

17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4

Answer

It is given that A = {1, 2, 3}.
The smallest equivalence relation containing (1, 2) is given by,
R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
Now, we are left with only four pairs i.e., (2, 3), (3, 2), (1, 3), and (3, 1).
If we odd any one pair [say (2, 3)] to R1, then for symmetry we must add (3, 2).
Also, for transitivity we are required to add (1, 3) and (3, 1).
Hence, the only equivalence relation (bigger than R1) is the universal relation.
This shows that the total number of equivalence relations containing (1, 2) is two.

The correct answer is B.

Page No. 31

18. Let f : R → R be the Signum Function defined as
and g : R → R be the Greatest Integer Function given by g (x) = [x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1]?

Answer

It is given that,
Also, g: R → R is defined as g(x) = [x], where [x] is the greatest integer less than or equal to x.
Now, let x (0, 1].
Then, we have
[x] = 1 if x = 1 and [x] = 0 if 0 < x < 1.

19. Number of binary operations on the set {a, b} are
(A) 10 
(B) 16 
(C) 20
(D ) 8

Answer

A binary operation * on {a, b} is a function from {a, b} × {a, b} → {a, b}
i.e., * is a function from {(a, a), (a, b), (b, a), (b, b)} → {a, b}.
Hence, the total number of binary operations on the set {a, b} is 24 i.e., 16.

The correct answer is B.

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