RELATNS-FNCTNSdoc-SS1-2012-13

7/31/2019 RELATNS-FNCTNSdoc-SS1-2012-13

1/5

GYAN BHARATI SCHOOL, SAKET, NEW DELHI

MATHEMATICS RELATION AND FUNCTIONS

CLASS - SS1___________________________________________________________________________________

1. Find x and y, if (x + 3.5) = (6, 2x + y).

2. If A = {1, 2, 3}, B = {3, 4} and C = {1, 3, 5}, Find

I). A x (B C) ii). A x (B C) iii). (A x B) (A x C)

3. If A x B = {(a, 1), (a, 5) ), (a, 2) ), (b, 2) ), (b, 5) ), (b, 1)}, find B x A.

4. If A = {1, 2}, form the set A x A x A.

5. Express A = {(a, b) : 2a + b = 5, a, bW } as the set of ordered pairs.Write its domain and range .

6. If A x B = {(a, 1), (b, 3) ), (a, 3) ), (b, 1) ), (a, 2) ), (b, 2)}, find A and B.

7. Let A and B be two sets such that A x B consists of 6 elements. If three elements of A x B are : (1, 4), (2, 6), (3, 6).

Find A x B and B x A.

8. If A = {1, 3, 5}, B = {x, y} , represent the following products by arrow diagrams :

i. A x B ii. B x A iii. A x A iv. B x B

9. Let A be the set of first ten natural numbers and let R be a relation on A defined by (x, y) R x + 2y = 10,

i.e., R = {(x, y) : x A, y A and x + 2y = 10}. Express R and R1 as sets of ordered pairs. Determine also

i. domains of R and R1 ii. ranges of R and R1 .

10. If A = {1, 2, 3}, B = {4, 5, 6}, which of the following are relations from A to B ? Give reasons in support of your answer.

i. R1 = {(1,4), (1,5), (1, 6)} ii. R2 = {(1,5), (2,4), (3, 6)}

iii. R3 = {(1,4), (1,5), (3, 6), (2, 6), (3, 4)} iv. R4 = {(4, 2), (2, 6), (5, 1), (2, 4)}

11. A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows : (x, y) R x divides y.

Express R as a set of ordered pairs and determine the domain and range of R. Also find R1 .

12. If R is the relation "less than" from A = {1, 2, 3, 4, 5} to B = {1, 4, 5}, write down the set of ordered pairs corresponding

to R. Find the inverse of R.

13. A relation R is defined on the set Z of integers as follows : (x, y) R x2 + y2 = 25. Express R and R1 as the sets of

ordered pairs and hence find their respective domains.

14. Let R be the relation on the set N of natural numbers defined by R = {(a, b) : a + 3b = 12, a N, b N}. Find :

i. R ii. Domain of R iii. Range of R

15. A relation R is defined from a set of A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows :(x, y) R x is relatively prime to y

Express R as a set of ordered pairs and determine its domain and range.

16. Find the inverse relation in each of the following cases :


2/5

i. R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

ii. R = {(x, y) : x, y N, x + 2y = 8}

iii. R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x 3.

17. Write the following relations as sets of ordered pairs :i. A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.

ii. A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y) R x is relatively prime to y.

18. Determine the domain and range of the relation R defined by

i. R = {(x, x + 5) : x {0, 1, 2, 3, 4, 5}}

ii. R = {(x, x3) : x is a prime number less than 10}

19. Let R be a relation from N to N defined by R = {(a, b) : a, b N and a = b2}. Are the following statements true ?

i. (a, a) R for all a N ii. (a, b) R (b, a) R

iii. (a, b) R and (b, c) R (a, c) R

20. Let A = {1, 2, 3, ......., 14}. Define a relation on a set A by R = {(x, y) : 3x y = 0, where x, y A}. Depict this

relationship using an arrow diagram. Write down its domain, co domain and range.

21. The adjacent figure shows a relationship between the sets A and B. Write this relation in (i) set builder form

(ii) roster form. What is its domain and range ?

22. If n (A) = 3, n (b) = 4, then write n (A x A x B).

23. If R is a relation defined on the set Z of integers by the rule (x, y) R x 2 + y2 = 9, then write the domain of R.

24. Let A = {1, 2, 3} and R = {(a, b) : |a2 b2| 5, a , b A}. then write R as set of ordered pairs.

25. Let R = {(x, y) : x, y Z, y = 2x 4}. If (a, 2) and (4, b2) R, then write the value of a and b.

26. If R = {(2, 1), (4, 7), (1, 2), ....}, then write the linear relation between the components of the ordered pairs of the

relation R.


3/5

27. If A = {1, 3, 5} and B = {2, 4}, list the elements of R, if R = {(x, y) : x, y A x B and x > y}.

28. If R = {(x, y) : x, yW, 2x + y = 8}, then write the domain and range of R.

29. Let A and B be two sets such that n (A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, write A and B.

30. Let A = {1, 2, 3, 5}, B = {4, 6, 9} and R be a relation from A to B defined by R = {(x, y) : x y is odd}. Write R in rosterform.

31. A relation from C to R is defined by x y | x | = y. Which one is correct ?

i. (2 + 3i) 13 ii. 3 (3) iii. (1 + i) 2 iv. i 1

32. Let A = {2, 1, 0, 1, 2} and f : A Z given by f (x) = x 2 2x 3. Find

i. the range of f ii. pre images of 6, 3 and 5

33. Express the following functions as sets of ordered pairs and determine their ranges :

i. f : A R, f (x) = x2 + 1, where A = {1, 0, 2, 4}

ii. g : A N, g(x) = 2x, where A = {x : x N, x 10}

34. Find the domain for which the functions f (x) = 2x2 1 and g (x) = 1 3x are equal.

35. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function ? If this is described by the formula, g(x) = x + , then what values should

be assigned to and ?

36. Le f : R R be given by f(x) = x 2 + 3. find i. {x : f(x) = 28} ii. the pre-images of 39 and 2 under f.

37. If f : R R be defined as follows :

Find i. f(1/2), f (), f(2) ii. range of f iii. preimages of 1 and 1

38. Write the following relations as sets of ordered pairs and find which of them are functions :

i. {(x, y) : y = 3x, x {1, 2, 3}, y {3, 6, 9, 12}}

ii. {(x, y) : y > x + 1, x = 1, 2 and y = 2, 4, 6 }

iii. {(x, y) : x + y = 3, x, y {0, 1, 2, 3}}

39. Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}. Determine which of the following sets are functions given by

from X and Y

i. f1 = {(1, 1), (2, 11), (3, 1), (4, 15)}

ii. f2 = {(1, 1), (2, 7), (3, 5)}

iii. f3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

40. If f(x) = 3x4 5x2 + 9, find f (x 1).

41. If f (x) = x + 1/x, prove that [f(x)]3 = f(x3) + 3f(1/x).

42. If f(x) = 1/(2x + 1), x 1/2 , then show that


4/5

43. If then show that , provided that x 0.

44. Let f be defined by f(x) = x 4 and g be defined by

Find such that f(x) = g(x) for all x.

45. If f is a real function defined by , then prove that :

46. Find the domain of each of the following real valued functions :

i. ii. iii. iv.

47. Find the domain of each of the following functions :i. f(x) = (x 2) ii. f (x) = 1/ (1 x) iii. f (x) = (4 x2)

48. Find the domain and range of the function f (x) given by

49. Find the range of each of the following functions :

i. f (x) = 1/ (x 5) ii. f (x) = (16 x2) iii. f (x) = x/ (1 + x2) iv. f (x) = 3/ (2 x3)

50. Find the domain and range of the function .

51. Find the domain and range of the real valued function f (x) given by

52. Find the domain and range of the function :

53. Find the domain and range of the function .

===========================================================================================


5/5

RELATNS-FNCTNSdoc-SS1-2012-13

Documents

Transcript of RELATNS-FNCTNSdoc-SS1-2012-13