Solutions of chapter 1. Relations And Functions | Mathematics - Exemplar

1

Let A = {a, b, c} and the relation R be defined on A as follows:

R = {(a, a), (b, c), (a, b)}.

Then, write minimum number of ordered pairs to be added in R to make R reflexive and transitive.

2

Let D be the domain of the real valued function f defined by . Then, write D.

3

Let f, g :R → R be defined by f(x) = 2x + 1 and g (x) = x² – 2, i∀ x ∈ R, respectively. Then, find g o f.

4

Let f :R → R be the function defined by f (x) = 2x – 3 ∀ x ∈ R. write f^–1.

5

If A = {a, b, c, d} and the function f = {(a, b), (b, d), (c, a), (d, c)}, write f^–1.

6

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

7

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = α x + β, then what value should be assigned to α and β.

8

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

(i) {(x, y): x is a person, y is the mother of x}.

(ii){(a, b): a is a person, b is an ancestor of a}.

9

If the mappings f and g are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g.

10

Let C be the set of complex numbers. Prove that the mapping f :C → R given by f (z) = |z|, ∀ z ∈ C, is neither one-one nor onto.

11

Let the function f :R → R be defined by f (x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto.

12

Let X = {1, 2, 3} and Y = {4, 5}. Find whether the following subsets of X × Y are functions from X to Y or not.

(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)}

(ii) g = {(1, 4), (2, 4), (3, 4)}

(iii) h = {(1,4), (2, 5), (3, 5)}

(iv) k = {(1,4), (2, 5)}.

13

If functions f : A → B and g : B → A satisfy g o f = I_A, then show that f is one one and g is onto.

14

Let f : R → R be the function defined by . Then, find the range of f.

15

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈Z, aRb if and only if a – b is divisible by n. Show that R is an equivalence relation.

16

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric and transitive.

17

Let R be relation defined on the set of natural number N as follows:

R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive.

18

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

(a) an injective mapping from A to B

(b) a mapping from A to B which is not injective

(c) a mapping from B to A.

19

Give an example of a map

which is one-one but not onto

19

Give an example of a map

which is not one-one but onto

19

Give an example of a map

which is neither one-one nor onto.

20

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by ∀ x ∈ A . Then show that f is bijective.

21

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

21

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

g(x) = |x|

21

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x |x|

21

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x²

22

Each of the following defines a relation on N:

x is greater than y, x, y ∈N

Determine which of the above relations are reflexive, symmetric and transitive.

22

Each of the following defines a relation on N:

x + y = 10, x, y ∈N

Determine which of the above relations are reflexive, symmetric and transitive.

22

Each of the following defines a relation on N:

x y is square of an integer x, y ∈N

Determine which of the above relations are reflexive, symmetric and transitive.

22

Each of the following defines a relation on N:

x + 4y = 10 x, y ∈N.

Determine which of the above relations are reflexive, symmetric and transitive.

23

Let A = {1, 2, 3, ... 9} and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A ×A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

24

Using the definition, prove that the function f : A → B is invertible if and only if f is both one-one and onto.

25

Functions f, g : R → R are defined, respectively, by f (x) = x² + 3x + 1, g (x) = 2x – 3, find

(i) f o g (ii) g o f (iii) f o f (iv) g o g

26

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

(i) a * b = a – b ∀ a, b ∈Q

(ii) a * b = a² + b²∀ a, b ∈Q

(iii) a * b = a + ab ∀ a, b ∈Q

(iv) a * b = (a – b)²∀ a, b ∈Q

27

Let * be binary operation defined on R by a * b = 1 + ab, ∀a, b ∈R. Then the operation * is

(i) commutative but not associative

(ii) associative but not commutative

(iii) neither commutative nor associative

(iv) both commutative and associative

28

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀a, b ∈T. Then R is

29

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is

30

The maximum number of equivalence relations on the set A = {1, 2, 3} are

31

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

32

Let us define a relation R in R as aRb if a ≥ b. Then R is

33

Let A = {1, 2, 3} and consider the relation

R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.

Then R is

34

The identity element for the binary operation * defined on Q ~ {0} asb ∈Q ~ {0}is

35

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

36

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is

37

Let f :R → R be defined byThen f is

38

Let f :R → R be defined by f (x) = 3x² – 5 and g : R → R by. Then g o f is

39

Which of the following functions from Z into Z are bijections?

40

Let f :R → R be the functions defined by f (x) = x³ + 5. Then f^–1 (x) is

41

Let f : A → B and g : B → C be the bijective functions. Then (g o f)^–1 is

42

Let be defined by. Then

43

Let f : [0, 1] → [0, 1] be defined by

Then (f o f) x is

44

Let f : [2, ∞) → R be the function defined by f (x) = x²–4x+5, then the range of f is

45

Let f : N → R be the function defined byand g : Q → R be another function defined by g (x) = x + 2. Then (g o f)3/2 is

46

Let f :R → R be defined by

Then f (– 1) + f (2) + f (4) is

47

Let f :R → R be given by f (x) = tan x. Then f^–1 (1) is

48

Fill in the blanks in each of the

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

49

Fill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a² – b²| < 8}. Then R is given by _______.

50

Fill in the blanks in each of the

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______and f o g = ______.

51

Fill in the blanks in each of the

Let f :R → R be defined by. Then (f o f o f) (x) = _______

52

Fill in the blanks in each of the

If f (x) = {4 – (x–7)³}, then f^–1(x) = _______.

53