**Sets and Relations theory**

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**1. If every element of a set P is also an element of set Q then set P is a subset of Q then the set Q is known as**

(a) Set of P

(b) Sub set of P

(c) Super set of P

(d) Any of these.

Every subset is a subset of a super set

Hence Option (c) is the correct answer

**2. The empty set is also known as**

(a) Zero set

(b) Null set

(c) Void set

(d) (b) or (c).

Empty set is also known as null set or void set.

Hence Option (d) is the correct answer

**3. Inter section of two set A & B is known as**

Intersection of two set A & B is known as ()

Hence Option (b) is the correct answer

**4. If n(A) = n(B), for two finite set A & B is known**

(a) Singleton set

(b) Equal sets

(c) Equivalent sets

(d) None of these.

Two finite sets are called equivalent if they have same number of element. i.e. n (A) = n (B)

Hence Option (c) is the correct answer

**5. The objects of a set are called its**

(a) Universal set

(b) Sub set

(c) Unit

(d) Elements of set.

Objects of a set are called its element

Hence Option (d) is the correct answer

**6. The methods of describing a set**

(a) Roster from

(b) Set Builder from

(c) Either (a) or (b)

(d) None of the above

There are two method of describing a set

(1) Roster from

(2) Set builder from

Hence Option (c) is the correct answer

**7. Universal set is denoted by _____________ .**

(a) { }

(b)

(c) U

(d) None of these

Universal set is denoted by

Hence Option (c) is the correct answer

**8. _____________is also known as Tabular from.**

(a) Rule method

(b) Roster form

(c) Set-Builder form

(d) Any of the above

Another name for Roster from is Tabular form. In such form, the elements are being separated by commas and are enclosed within braces { }.

Hence Option (b) is the correct answer

**9. ____________ is also known as Rule Method.**

(a) Rule method

(b) Roster from

(c) Set-Builder from

(d) None.

Another name for Set Builder form is Rule Method. It is a statement or an expression to represent all the elements of a set.

Hence Option (c) is the correct answer

**10. Under _______________ , we just make a list of all elements of the set and put them under { }**

(a) Roster method

(b) Set-Builder form

(c) Both (a) & (b)

(d) None of the above.

Under Roster method we make a list of elements separated by comma and we put it under { }

Hence Option (a) is the correct answer

**11. .**

(a) Proper sub-set

(b) Equal set

(c) Equivalent sets

(d) Power set.

()

Hence Option (b) is the correct answer

**12. Set A and set B be two sets, the ______________of A and B is .**

(a) Intersection

(b) Equivalent

(c) Union

(d) Void

Intersection of set A & set B = ()

Hence Option (a) is the correct answer

**13. Set A = {1, 2, 3} & set B = {2, 1, 3}.**

(a) A & B are Equal

(b) A & B is disjoint

(c) A and B is NOT Equal

(d) A is proper sub set of B

Set A and set B are equal set because these sets have same elements.

Hence Option (a) is the correct answer

**14. If A = {2, 5, 7} and B = {3, 5, 7, 8}, the value of A B is**

(a) {5, 7, 8}

(b) {3, 8}

(c) {2, 3, 4, 5, 6, 7, 8}

(d) {0}

Set A = {2, 5, 7} B = {3, 5, 7, 8}

A B (A union B)

= {2, 3, 5, 7, 8}

Hence Option (c) is the correct answer

**15. If A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}, the value of A – {B C } is**

(a) {1, 2, 3}

(b) {2, 3, 4, 5}

(c) {1}

(d) {0}.

A = { 1, 2, 3, 4} B = {2, 4, 6, 8} C = {3, 4, 5, 6}

B C = {2, 3, 4, 5, 6, 8}

A – {B C } = {1, 2, 3, 4} – {2, 3, 4, 5, 6, 8} = {1}

Hence Option (c) is the correct answer

**16. Which of the following is an empty set ?**

(a) {0}

(b) {}

(c) 0

(d) .

{0} is not an empty set because 0 is an element.

{} is not an empty set because is an element.

is an empty set because it has no element

Hence Option (d) is the correct answer

**17. The subsets of {0} are**

(a) 1

(b) {0}

(c)

(d) and {0} .

Hence Option (d) is the correct answer

**18. If A = {1, 2, 3} and B = {2, 4}, the value of A B is**

(a) {1, 2, 4}

(b) {2}

(c) {1, 2}

(d)

Hence Option (b) is the correct answer

**19. , which one of the following statements is correct ?**

If A = then A . Because f is a subset of every set.

Hence Option (d) is the correct answer

**20. If A ={1, 2, 3}, B = {3, 4} and C = {4, 5, 6} then is equal to**

(a) {1, 2, 3,4}

(b) {3}

(c) {1, 2, 3, 4, 5, 6}

(d) {1, 2} .

Hence Option (a) is the correct answer

**21. Of the following sets, the one having exactly eight subsets is**

(a) {1, 2}

(b) {0, 1, 2}

(c) {8}

(d) {1, 2, 3, 4, 5, 6, 7, 8} .

The number of subsets (called Power Set) of a set of n elements contains 2^{n}-1 subsets. Here {0, 1, 2} has 3 elements. It contains 2^{3}-1 = 8 subsets :

Hence Option (b) is the correct answer

**22. If A = {1, 2, 3, 4} and B = {5, 6, 7}, then is**

A = {1, 2, 3, 4} B = {5, 6, 7}

(as there is no common elements.)

Hence Option (d) is the correct answer

**23. The number of subsets of the set {a , b, c} is**

(a) 12

(b) 6

(c) 8

(d) 9.

The number of subsets (called Power Set) of a set of n elements contains 2n-1 subsets.. Here {0, 1, 2} contains 23-1 = 8 subsets

Hence Option (c) is the correct answer

**24. {1, 2, 3} {2, 3, 4} is equal to**

(a) {1, 2}

(b)

(c) {1, 2}

(d) {1, 2, 3, 4} .

= {1, 2, 3} {2, 3, 4}

= {1, 2, 3, 4}

Hence Option (d) is the correct answer

**25. If A = {x : x**^{2}** ****–**** 3x ****+**** 2 = 0} and B = {x : x**^{2}** ****+**** 4x ****–**** 12 = 0}, then A ****–**** B is equal to**

Eq A : x2 – 3x + 2 = 0 Eq B : x2 + 4x – 12 = 0

Or, x2 – x – 2x + 2 = 0 Or, x2 + 6x – 2x – 12 = 0

Or, x (x – 1) – 2 (x – 1) = 0 Or, x (x + 6) – 2 (x + 6) = 0

Or, (x – 1) (x – 2) = 0 Or, (x – 2) (x + 6) = 0

Hence Option (c) is the correct answer

**26. Write down the power set of the set {0}**

(a)

(b) {0}

(c) {}

(d) {, {0} }.

Hence Option (d) is the correct answer

**27. If U = { 2, 3, 4, 5, 6, 7, 8, 9,10, 11}, A = {2, 4, 7}, B = {3, 5, 7, 9, 11} and C = {7, 8, 9, 10, 11}, compute : **

(a) {7}

(b)

(c) {2}

(d) {10}

U = { 2, 3, 4, 5, 6, 7, 8, 9,10, 11},

A = {2, 4, 7},

B = {3, 5, 7, 9, 11}

C = {7, 8, 9, 10, 11},

Hence Option (a) is the correct answer

**28. If U = n {a, b, c, d, e, f}, A = {a, b, c}, find (U A)**

(a) U

(b) A

(c)

(d) {a, b, c}

Hence Option (a) is the correct answer

**29. If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f} and C = {c, d, e}, find **

(a) {a, b, c}

(b) B

(c) {c, d, e}

(d) U .

A = {a, b, c}

B = {c, d, e, f}

C = {c, d, e}

Hence Option (d) is the correct answer

**30. If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find **

(a) {c}

(b) {a}

(c) {f}

(d) {e}

A = {a, b, c}

B = {c, d, e, f}

C = {c, d, e}

Hence Option (a) is the correct answer

**31. If S and T are two sets such that S has 21 elements, T has 32 elements, and S T has 11 elements, how many elements does S T have ?**

(a) 48

(b) 26

(c) 42

(d) 36

S has 21 elements

T has 32 elements

So, there are 11 common elements in S & T

So, uncommon elements in S = 21 – 11 = 10

Un common elements in T = 32 – 11 = 21

We may also calculate it as 21 (S) + 32 (T) – 11 common elements = 53-11=42

Hence Option (c) is the correct answer

**32. In a group of 1,000 people, there are 750 people who can speak Hindi and 400 who can speak English. How many can speak Hindi only ? **

(a) 600

(b) 700

(c) 850

(d) 900

No. of people can speak in Hindi = 750

No. of people can speak in English = 400

No. of people can speak in both Hindi & English

= 750 + 400 – 1000 = 1150 – 1000 = 150

No. of people can speak in only Hindi = 750 – 150 = 600

Hence Option (a) is the correct answer

**33. In a class of 50 students, 35 opted for Mathematics and 37 opted for Biology. How many have opted for both Mathematics and Biology ? How many have opted for only Mathematics ?**

**(Assume that each student has to opt for at least one of the subjects).**

(a) 20

(b) 21

(c) 13

(d) 18

No. of students opted for Mathematics = 35

No. of students opted for Biology = 37

No. of students opted for Mathematics & Biology

= 35 + 37 – 50 = 72 – 50 = 22

No. of students opted only for Mathematic (35 – 22) = 13

Hence Option (c) is the correct answer

**34. In a town with a population of 5,000, 3,200 people are egg-eaters, 2,500 meat-eaters and 1,500 eat both egg and meat. How many are pure vegetarians ?**

(a) 400

(b) 800

(c) 700

(d) 750

Both egg-eater & meat-eater = 1500

Only egg-eater = 3200 – 1500 = 1700

Only meat-eater = 2500 – 1500 = 1000

vegetarians = 5000 – (1700 + 1500 + 100) = 5000 – 4200 = 800

Hence Option (b) is the correct answer

**35. If S = {x : x**^{2}** ****+**** 1 = 0, x real}, then S is**

Hence Option (d) is the correct answer

**36. Which of the following is correct statement ?**

Hence Option (a) is the correct answer

**37. If A = {1, 2, 3}, B = {2, 3, 4}, then A B is**

(a) {2, 3}

(b) {2}

(c) {3,2}

(d) Both a & c

Hence Option (d) is the correct answer

**38. Let A = {0, 1, 3, 4}, B = {5, 6, 1, 3, 9} and C = {1, 2, 3, 4, 9, 13}, then is**

(a) C

(b) {0, 1, 2, 3, 4, 9, 13}

(c) A

(d) B

A = {0, 1, 3, 4}

B = {5, 6, 1, 3, 9}

C = {1, 2, 3, 4, 9, 13}

Hence Option (a) is the correct answer

**39. Let A = {1, 2, 3,4,5,6,7,8,9}, B = {2, 4, 6, 7, 8} and C = {3, 4, 5, 8, 9, 10}, then (A ****–**** B)**

(a) {1, 3, 4, 5, 8, 9, 10}

(b) {1, 2, 3, 4, 5, 9}

(c) {6, 7, 8}

(d) {1, 3, 4}

A = {1, 2, 3, 4,5,6,7,8,9},

B = {2, 4, 6, 7, 8}

C = {3, 4, 5, 8, 9, 10},

A – B = {1, 3, 5, 9}

Hence Option (a) is the correct answer

**40. Which one of the following is a null set ?**

(a) {x : x + 1 = 1}

(c) {x : x > 1 and x < 1}

Hence Option (c) is the correct answer

**41. In a group of 75 persons, 20 take tea but not coffee and 30 take tea. How many take coffee but not tea ?**

(a) 45

(b) 32

(c) 50

(d) 60

Only tea = 20

both coffee & tea = 30 – 20 = 10

Only coffee = 75 – (20 + 10) = 45

Hence Option (a) is the correct answer

**42. Set the positive integers, less than 3 is**

(a) {-1}

(b) {0, 1}

(c) {1, 2, 3}

(d) {1, 2}

Positive integers less 3 are 1, 2.

So, answer = {1, 2}

Hence Option (d) is the correct answer

**43. A = {2, 3, 5, 7}, B = {4, 6, 8, 10} then A B can be written as**

Hence Option (a) is the correct answer

**44. By Roster method to express integers greater than 5 and less than 8.**

Integers greater than 5 and less than or equal to 8 are 6, 7,

So, required answer {6, 7}

Hence Option (d) is the correct answer

**45. Write the set containing all days of the week beginning with S.**

(a) {Sunday, Monday}

(b) {Saturday, Sunday}

(c) {Friday, Saturday}

(d) none

Days Monday , Wednesday, Tuesday, Thursday, Friday, Saturday, Sunday

So, all days of the week beginning with S are Saturday, Sunday.

Required set = {Saturday, Sunday}

Hence Option (b) is the correct answer

**46. A **

**A is equal to**

(a) A

(b) U

(c)

(d) A’

Hence Option (a) is the correct answer

**47. A **

**A’ is equal to**

(a) U

(b)

(c) A

(d) A A’

Hence Option (b) is the correct answer

**48. A = {1, 2, 3, 4, 5}, B = {4, 6, 7, 5}, C = {2, 3, 8, 9}, then is**

(a) Set A

(b) Set B

(c) Set C

(d) {2, 3, 4, 5}

Hence Option (d) is the correct answer

**49. V = {x : x is vowel}, Y = {x : x is ‘a’ or any letter before ‘e’ in the alphabet}, then V Y is**

(a) {a, o, u}

(b)

(c) {a, e, i, o, u}

(d) {a}

Hence Option (d) is the correct answer

**50. If A has 32 elements, B has 42 elements and A ** **B has 62 elements. The number of elements in** **A B is**

(a) 12

(b) 70

(c) 20

(d) 50

Hence Option (a) is the correct answer

**51. In a group of 20 children, 8 drink tea but not coffee and 13 like tea. The number of children drinking coffee but not tea is**

(a) 5

(b) 7

(c) 3

(d) 4.

Total number of children = 20

Drink only tea = 8

Like tea = 13

Drink both tea & coffee (13 – 8) = 5

Drink only coffee = (20 – 8 – 5) = 7

Hence Option (b) is the correct answer

**52. The number of subsets of the {6, 8, 11} is**

(a) 7

(b) 5

(c) 8

(d) 3

Number of subsets = 2n-1, where n is the number of elements in the given set. The number of elements in the set is 3. So, the number of subsets = 23 -1 =8, having elements as :

Hence Option (c) is the correct answer

**53. If A = {a, b, c, d, e, f} B= {a, e, i, o, u} and C = {m, n, o, p, q, r, s, t, u} then A B is**

Hence Option (a) is the correct answer

**54. On a survey of 100 boys it was found that 50 used white shirt, 40 used red and 30 used blue. 20 were habituated in using both white and red shirts, while 15 used both red and blue shirts and 10 used blue and white shirts. Find the number of boys using all the colours.**

(a) 22

(b) 25

(c) 45

(d) 40.

Hence Option (b) is the correct answer