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PACE BATCH (11th Commerce)

SETS Practice Sheet

1. Which of the following collections are sets? Justify 6. Write the following intervals in set builder form:
your answer: (i) (-7, 0) (ii) [6, 12]
(i) A collection of all natural numbers less than (iii) (6, 12) (iv) [-20, 3]
50.
(ii) The collection of good hockey players in India.
(iii) The collection of all girls in your class. 7. Let A = {∅, {∅}, 1, {1, ∅}, 2} which of the following
(iv) The collection of difficult topics in are true?
mathematics. (i) ∅∈A (ii) {∅} ← A
(v) The collection of all questions in this chapter. (iii) {1} 𝜖 A (iv) {2, ∅} C A
(vi) A collection of most dangerous animals of the (v) 2CA (vi) {2, {1}} C/A
world.
(vii) The collection of prime integers. (vii) {{2}, {1}} C/A
(viii) { ∅, {∅}, {1, ∅}} C A
2. Describe each of the following sets in Roster form: (ix) {{∅}} C A
(i) {x : x ϵ z and |x| ≤ 2 }
(ii) {𝑥 ∶ 𝑥 ∈ 𝑧, 𝑥 2 < 20 }
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8. Let U = {1, 2, 3, 4, 5, 6, 7, 8 , 9}, A = {1, 2, 3, 4},
(iii) {𝑥 ∶ 𝑥 = 2𝑛 − 1
,1 ≤ 𝑛 ≤ 5} B = {2, 4, 6, 8} , C = {3, 4, 5, 6} Prove that :
(iv) {𝑥 ∶ 𝑥 is a letter of the word MISSISSIPPI} (i) (A U B)’ = A’ ∩ B’
𝑛
(v) {𝑥: 𝑛2 + 1
and 1 ≤ 𝑛 ≤ 3, 𝑛𝜖𝑁} (ii) (A ∩ B)’ = A’ U B’

3. Describe each of the following sets in set – builder 9. Let A = {1, 2, 3, , 4, 5}, B = {2, 3, 5, 6}, C = {4, 5, 6,
form: 7} . Verify the following identities:
1 1 1 1
i. {1, 4 , 9 , 16 , 25 , … } (i) A ∪ (B ∪ C) = (A U B) ∩ (A ∪ C)
ii. {
1
1 ,2
1
,3
1
,4
1
,5…} (ii) A ∩ (B – C) = (A ∩ B) – (A ∩ C)
(iii) A – (B ∩ C) = (A – B) U (A – C)
iii. {1, 2, 3, 4, 5, 6}
iv. {0, 3, 6, 9, 12 …}
10. Show that for any sets A and B,
4. Find the pairs of equal sets, from the following sets if A ∪ (B – A) = A ∪ B
any, giving reasons: A = {0},
B = {𝑥 ∶ 𝑥 > 15 𝑎𝑛𝑑 𝑥 < 5} ,
11. If X and Y are two sets such that n(X) = 17, n(Y) =
C = {𝑥 ∶ 𝑥 − 5 = 0 }, D = {𝑥 ∶ 𝑥 2 = 25} , 23 and n(X ∪ Y) = 38, find n(X ∩ Y).
E = {x: x is an integral positive root of the equation
x2 – 2x – 15 = 0}
12. Out of 500 car owner investigated, 400 owned Maruti
5. Write the following subset of R as intervals: car and 200 owned Hyundai car; 50 owned both
cars. Is this data correct?
(i) {𝑥 ∶ 𝑥 ∈ 𝑅, −4 < 𝑥 ≤ 6}
(ii) {𝑥 ∶ 𝑥 ∈ 𝑅, −12 < 𝑥 < −10} 13. If A, B and C are three sets and U is the universal set
(iii) {𝑥 ∶ 𝑥 ∈ 𝑅, 0 ≤ 𝑥 < 7} such that n(∪) = 700, n(A) = 200, n(B) = 300 and n
(A ∩ B) = 100. Find n(A’ ∩ B’).
(iv) {𝑥 ∶ 𝑥 ∈ 𝑅, 3 ≤ 𝑥 ≤ 4}
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14. In a class of 35 students, 17 have taken mathematics, 19. There are 40 students in a chemistry class and 60
10 have taken mathematics but not economics. Find students in a physics class. Find the number of
the no. of students who have taken both mathematics students which are either in physics class or
and economics and the no. of students who have chemistry class in the following cases:
taken economics but not mathematics, if it is given (i) the two classes meet at the same hour.
that each student has taken either mathematics or (ii) the two classes meet at different hours and 20
economics or both. students are enrolled in both the subjects.

15. In a survey of 25 students, it was found that 15 had 20. If A and B be two sets containing 3 and 6 elements
taken Mathematics, 12 had taken Physics and 11 had respectively, what can be the minimum number of
taken Chemistry, 5 had taken Mathematics and elements in A  B? Find also, the maximum number
Chemistry, 9 had taken Mathematics and Physics, 4
of elements in A  B.
had taken Physics and Chemistry and 3 had taken all
the three subjects. Find the no. of students that had:
(i) Only Chemistry 21. Write the following sets in the roaster form.
(ii) Only Mathematics (i) A = {x : x ∈ R, 2x + 11 = 15}
(iii) Only Physics (ii) B = {x : x2 = x, x ∈ R}
(iv) Physics and Chemistry but not Mathematics (iii) C = {x : x is a positive factor of a prime number
p}
(v) Mathematics and Physics
(vi) Only one of the subjects but not Chemistry.
22. Given that N = {1, 2, 3,...,100}. Then, write.
(vii) At least one of the three subjects
(i) the subset of N whose elements are even
(viii) None of the subjects
numbers.
(ii) the subset of N whose elements are perfect
16. A survey of 500 television viewers produced the square numbers.
following information, 285 watch football, 195 watch
hockey, 115 watch basketball, 45 watch football and
basketball, 70 watch football and hockey, 50 watch 23. Out of 100 students; 15 passed in English, 12 passed
hockey and basketball, 50 do not watch any of the in Mathematics, 8 in Science, 6 in English and
three games. How many watches all the three games? Mathematics, 7 in Mathematics and Science, 4 in
English and Science, 4 in all the three. Find how
many passed
17. In a survey it was found that 21 persons liked product
(i) in English and Mathematics but not in Science.
P1, 26 liked product P2 and 29 liked product P3. If 14
persons liked products P1 and P2; 12 persons liked (ii) in Mathematics and Science but not in English.
product P3 and P1; 14 persons liked products P2 and (iii) in Mathematics only.
P3 and 8 liked all the three products. Find how many (iv) in more than one subject only.
liked product P3 only
24. In a group of 50 students, the number of students
18. In a survey of 60 people, it was found that 25 people studying French, English, Sanskrit were found to be
read newspaper H, 26 read newspaper T, 26 read as follows French = 17, English = 13, Sanskrit = 15
newspaper I,9 read both H and I, 11 read both H and French and English = 09, English and Sanskrit = 4,
T, 8 read both T and I, 3 read all three newspapers. French and Sanskrit = 5, English, French and Sanskrit
Find: = 3. Find the number of students who study
(i) the numbers of people who read at least one of (i) only French.
the newspapers. (ii) only English.
(ii) the number of people who read exactly one (iii) only Sanskrit.
newspaper. (iv) English and Sanskrit but not French.
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(v) French and Sanskrit but not English. 27. In a town of 840 persons, 450 persons read Hindi, 300
(vi) French and English but not Sanskrit. read English and 200 read both. Then, the number of
(vii) atleast one of the three languages. persons who read neither, is
(viii) none of the three languages (1) 210 (2) 290
(3) 180 (4) 260
25. Two finite sets have m and n elements. The number
of subsets of the first set is 112 more than that of the 28. If X = {8n - 7n - 1 | n ∈ N} and y = {49n - 49 | n ∈
second set. The values of m and n are, respectively N}. Then,
(1) 4, 7 (2) 7, 4 (1) X⊂Y (2) YcX
(3) 4, 4 (4) 7, 7 (3) X=Y (4) X∩Y=ϕ

26. Let F1 be the set of parallelograms, F2 the set of 29. A survey shows that 63% of the people watch a news
rectangles, F3 the set of rhombuses, F4 the set of channel whereas 76% watch another channel. If x%
squares and F5 the set of trapeziums in a plane. Then, of the people watch both channel, then
F1 may be equal to (1) x = 35 (2) x = 63
(1) F2 ∩ F3 (2) F3 ∩ F4 (3) 39 ≤ x ≤ 63 (4) x = 39
(3) F2 ∪ F5 (4) F2 ∪ F3 ∪ F4 ∪ F1
30. If A and B are two sets, then A ∩ (A ∪ B) equals to
(1) A (2) B
(3) ϕ (4) A∩B
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Answer Key
1. (H & S) 16. (H & S)

2. (H & S) 17. (H & S)

3. (H & S) 18. (H & S)

4. (H & S) 19. (H & S)

5. (H & S) 20. (H & S)

6. (H & S) 21. (H & S)

7. (H & S) 22. (H & S)

8. (H & S) 23. (H & S)

9. (H & S) 24. (H & S)

10. (H & S) 25. (H & S)

11. (H & S) 26. (H & S)

12. (H & S) 27. (H & S)

13. (H & S) 28. (H & S)

14. (H & S) 29. (H & S)

15. (H & S) 30. (H & S)

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Hints and Solutions

1. (i), (iii), (v), (vii) 11. (2)

2. i. {–2, –1, 0, 1, 2} 12. Data is incorrect

ii. {–4, –3, –2, –1, 0, 1, 2, 3, 4}
iii.
1 1 1 1 13. 300
{1 , 3
,5 ,7 ,9 }
iv. {M, I, S, P}
14. 18
1 2 3
v. {2 , 5 , 10 }

15. (i) 5 (ii) 4 (iii) 2 (iv) 1

1
3. i. {𝑥 ∶ 𝑥 = , 𝑛 ∈ 𝑁} (v) 6 (vi) 11 (vii) 23 (viii) 2
𝑛2
1
ii. {𝑥 ∶ 𝑥 = 𝑛 , 𝑛 ∈ 𝑁}
16. 20, 325
iii. {𝑥 ∶ 𝑥 ∈ 𝑁, 𝑥 < 7}
iv. {𝑥 ∶ 𝑥 = 3𝑛, 𝑛 ∈ 𝑧} 17. 11

4. C=E 18. 43

5. (i) (–4, 6] 19. (i) 100 (ii) 80

(ii) (–12, –10)
(iii) [0, 7) 20. 6
(iv) [3, 4]

21. i. {2} ii. {0,1} iii. {1,p}

6. (i) {𝑥 ∶ 𝑥 ∈ R and − 7 < 𝑥 < 0}
(ii) {𝑥 ∶ 𝑥 ∈ R and 6 ≤ 𝑥 ≤ 12} 22. (i) Required subset = {2, 4, 6, 8,..., 100}
(iii) (6, 12] = {𝑥 ∶ 𝑥 ∈ R and 6 < 𝑥 ≤ 3} (ii) Required subset = {1, 4, 9,16,25, 36, 49,64,
(iv) {𝑥 ∶ 𝑥 ∈ R and − 20 < 𝑥 < 3} 81,100}

7. (i) T (ii) T 23. i. 2 ii. 3 iii. 3 iv. 9

(iii) F (iv) T
(v) F (vi) T 24. (i) Number of students who study French only, a
(vii) T (viii) T =6
(ix) T (ii) Number of students who study English only, c
=3
(iii) Number of students who study Sanskrit only, g
8. With discussion
=9
(iv) Number of students who study English and
9. With discussion
Sanskrit but not French, d = 1
(v) Number of students who study French and
10. With discussion Sanskrit but not English, f = 2
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(vi) Number of students who study French and 26. (4)

English but not Sanskrit, b = 6
(vii) Number of students who study atleast one of 27. (2)
the three languages
=a+b+c+d+e+f+g= 6+ 6+ 3+1+3 + 2 + 9=30
28. (1)
(viii) Number of students who study none of three
languages = Total students – Students who
study atleast one of the three languages 29. (3)

= 50 - 30 = 20
25. (1) 30. (1)

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