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Sets Problems with solutions for Class 11 Maths





Fill in the blank


(a) If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, Then A ∪ B is _________
(b) The sets A and B are having elements 10 and 8,n(A ∩ B) =2, the  n(A ∪B) is _______
(c) A set Z contains 4 elements, and then the number of elements in the Power set of Z will be ______
(d) The set Z={x: x2 -3=0,x is a rational number) is an______set

True or False statement

(1) Set of odd natural numbers divisible by 2 is a null set
(2) Set of even prime numbers is not an null set
(3) {x:x is a natural numbers, x < 4 and x > 11 } is an infinite set
(4) {y:y is a point common to any two parallel lines} is an infinite set
(5) The set of months of a year is a finite set
(6) {0,1, 2, 3 ...} is a finite set
(7) {1, 2, 3 ... 999} is an infinite set
(8) The set of positive integers greater than 99 is an infinite set
(9) The set of lines which are parallel to the y-axis  is an infinite set
(10) The set of letters in the English alphabet is a finite set
(11) The set of natural numbers under 200 which are multiple of 7 is finite set
(12) The set of animals living on the earth is a finite set
(13) The set of circles passing through the origin (0, 0) is a finite set
(14) The set A = {-2, -3}; B = {x: x is solution of x2 + 5x + 6 = 0} are equal sets
(15) The set P = {x: x is a letter in the word FOLLOW}; Q = {y: y is a letter in the word WOLF} are not equal sets
(16) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
(17) {a, b, c}⊄ {b, c, d}
(18) {x: x is a student of Class X of your school} ⊄ {x: x student of your school}
(19) {x: x is a square in the plane} ⊄ {x: x is a rectangle in the same plane}
(20) {p: p is a triangle in a plane} ⊂ {p: p is a rectangle in the plane}
(21) {x: x is an equilateral triangle in a plane} ⊂ {x: x is a triangle in the same plane}
(22) {y: y is an odd natural number} ⊂ {y: y is an integer}
(23) {a, b} ⊄ {b, c, a}
(24) {a, e} ⊂ {p: p is a vowel in the English alphabet}
(25) {1, 2, 3} ⊂{1, 3, 5}
(26) {p} ⊂ {p, q, s}
(27) {a} ∈ (1, 2, 3)
(28) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}
 (29) If x ∈ A and A ∈ B, then x ∈ B
(30) If A ⊂ B and B ∈ C, then A ∈ C
(31) If A ⊂ B and B ⊂ C, then A ⊂ C
(32) If A ⊄ B and B ⊄ C, then A ⊄ C
(33) If x ∈ A and A ⊄ B, then x ∈ B
(34) If A ⊂ B and x ∉ B, then x ∉ A


Solutions
  1. T
  2. T
  3. F
  4. F
  5. T
  6. F
  7. F
  8. T
  9. T
  10. T
  11. T
  12. T
  13. F
  14. T
  15. F
  16. T
  17. T
  18. F
  19. F
  20. F
  21. T
  22. T
  23. F
  24. T
  25. F
  26. T
  27. F
  28. T
  29. F
  30. F
  31. T
  32. F
  33. F
  34. T

Subjective Questions


Write the following sets in roster form:
(1) U = {x: x is an integer and –10 < x < 10}.
(2) V = {x: x is a natural number less than 18}.
(3) W = {x: x is a two-digit natural number such that the sum of its digits is 9}
(4) X = {x: x is a prime number which is divisor of 90}.
(5) Y= The set of all letters in the word MATHEMATICS.
(6) Z= The set of all letters in the word INTEGRATION.

Linked Type comprehension


1. If U = {1,2,3, 5, 7, 9, 11}, V = {7, 9, 11, 13}, W = {11, 13, 15} and X = {15, 17,19,21,23}; find
(i) U ∩ V
(ii) V ∩ W
(iii) U ∩ W ∩ X
(iv) U ∩ W
(v) V ∩ X
(vi) U ∩ (V ∪ W)
(vii) U ∩ (V ∪ X)
(viii) (U ∩ V) ∩ (V ∪ W)
(ix) (U ∪ X) ∩ (V ∪ W)
2.If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B – C
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D

Subjective Questions

1. In a group of 400 people in USA, 250 can speak Spanish and 200 can speak English. How many people can speak both Spanish and English?
Solution
Let S be the set of people who speak Spanish, and
E be the set of people who speak English
n(S ∪ E) = 400, n(S) = 250, n(E) = 200
n(S ∩ E) = ?
We know that:
n(S ∪ E) = n(S) + n(E) – n(S ∩ E)
∴ 400 = 250 + 200 – n(S ∩ E)
⇒ 400 = 450 – n(S ∩ E)
n(S ∩ E) = 450 – 400
n(S ∩ E) = 50
Thus, 50 people can speak both Spanish and English.
2. If P and Q are two sets such that P has 40 elements, P ∪Q has 60 elements and P ∩Q has 10 elements, how many elements does Y have?
Solution
It is given that:
n(P) = 40, n(P ∪ Q) = 60, n(P ∩ Q) = 10
We know that:
n(P ∪ Q) = n(P) + n(Q) – n(P ∩ Q)
∴ 60 = 40 + n(Q) – 10
n(Q) = 60 – (40 – 10) = 30
Thus, the set Q has 30 elements.

Subjective question on number of elements

(a) U={x: x is positive integer less than 1000 and divisible by 7}  , n(U) =?
(b) V={x: x is positive integer less than 1000 and divisible by 7 but not by 11} , n(V)=?
(c) P={x: x is positive integer less than 1000 and divisible by 7 and  11} , n(P)=?
(d) Q={x: x is positive integer less than 1000 and divisible by either 7 or  11} , n(Q)=?
(e) A={x: x is positive integer less than 1000 and divisible by exactly one of 7 or 11} , n(A)=?
(f ) B={x: x is positive integer less than 1000 and divisible by neither 7 nor 11} , n(B)=?
(g) C={x: x is positive integer less than 1000 and have distinct digits} , n(C)=?
(h) D={x: x is positive integer less than 1000 and have distinct digits and even} , n(D)=?
Solutions
  1. 142
  2. 130
  3. 12
  4. 220
  5. 208
  6. 779
  7. 738
  8. 373

Multiple Choice Questions

1.If X and Y are two sets and X′ denotes the complement of X, then X ∩ (X ∪ Y)′ is equal to (A) X (B) Y (C) φ (D) X ∩ Y 2.For all sets A and B, A – (A ∪ B) is equals to? (A) A ∩ B (B) A ∪ Β′ (C) A - B (D) None of the above 3.Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are. (A) 6,7 (B) 6,3 (C) 3,6 (D) 7,6 Solutions 1. C 2. B 3. B

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