Class 11 Mathematics Sets MCQs Set 02

SET THEORY

• \(n(A \Delta B)\) = number of elements which belong to exactly one of A or B.
  \(n(A \Delta B) = n\{(A - B) \cup (B - A)\}\)
  \(= n(A) + n(B) - 2n(A \cap B)\)
  \(= n(A \cup B) - n(A \cap B)\)
• A and B are two sets and \(n(A) = p\), \(n(B) = q\), Then
  (i) \(\min\{n(A \cup B)\} = \max\{p, q\}\)
  (ii) \(\max\{n(A \cup B)\} = p + q\)
  (iii) \(\min\{n(A \cap B)\} = 0\)
  (iv) \(\max\{n(A \cap B)\} = \min\{p, q\}\)
• No. of elements in exactly one of the sets A, B, C
  \(= n(A) + n(B) + n(C) - 2n(A \cap B) - 2n(B \cap C) - 2n(A \cap C) + 3n(A \cap B \cap C)\)
• No. of elements in exactly two of the sets A, B, C
  \(= n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C)\)

Question. Which of the following is an empty set
(a) \(\{\phi\}\)
(b) \(\{0\}\)
(c) \(\{n \in N \text{ and } n < 1\}\)
(d) The set of all even prime numbers
Answer: (c) \(\{n \in N \text{ and } n < 1\}\)
Hint: N = {1, 2, 3, ..............}

Question. Let A and B be two sets such that \(A \cup B = A\)
Then \(A \cap B\) is equal to
(a) \(\phi\)
(b) B
(c) A
(d) \(A \cup B\)
Answer: (b) B
Hint: Since, \(A \cup B = A \implies B \subseteq A \therefore A \cap B = B\)

Question. Which of the following is not correct?
(a) \(A \subseteq A^c\) if and only if \(A = \phi\)
(b) \(A^c \subseteq A\) if and only if A = X, where X is the universal set
(c) If \(A \cup B = A \cup C\), then B = C
(d) A = B is equivalent to \(A \cup C = B \cup C\) and \(A \cap C = B \cap C\)
Answer: (c) If \(A \cup B = A \cup C\), then B = C
Hint: \(A^c\) satisfies (1) and (2) by definition, (4) also follows trivially.
Assuming A to be any set other than the empty set, also B = A and \(C = \phi\), we have
\(A \cup B = A = A \cup C\)
But \(B \neq C\), So (3) is incorrect

Question. If A and B are two sets then \((A - B) \cup (B - A) \cup (A \cap B)\) is
(a) \(A \cup B\)
(b) \(A \cap B\)
(c) A
(d) \(B^1\)
Answer: (a) \(A \cup B\)
Hint: \((A - B) \cup (B - A) \cup (A \cap B) = A \cup B\)
Draw venn diagram

Question. \(A \cap (A \cup B)^C =\)
(a) A
(b) B
(c) \(\phi\)
(d) A-B
Answer: (c) \(\phi\)
Hint: \(A \cap (A \cup B)^C = A \cap (A^C \cap B^C) = (A \cap A^C) \cap (A \cap B^C) = \phi \cap (A \cap B^C) = \phi\)

Question. If \(U = \{a, b, c, d, e, f, g, h\}\) and \(A = \{a, b, c\}\) then complement of A is
(a) \(\{d, e, f\}\)
(b) \(\{d, e, f, g, h\}\)
(c) \(\{a, b, c\}\)
(d) \(\{d, e, g, h\}\)
Answer: (b) \(\{d, e, f, g, h\}\)
Hint: Complement of A = U - A

Question. Which of the following not a well defined collection of objects
(a) The set of Natural Numbers
(b) Rivers of India
(c) Various kinds of Triangles
(d) Five most renowned Mathematicians of the world.
Answer: (d) Five most renowned Mathematicians of the world.
Hint: For determining a mathematicians most renowned may vary from person to person

Question. Write the solution set of the equation \(x^2 + x - 6 = 0\) in roster form
(a) {2, -3}
(b) {-1, -2}
(c) {1, 2}
(d) {-1, 2}
Answer: (a) {2, -3}
Hint: \(x^2 + x - 6 = 0 \implies x = 2, -3\)

Question. Write the set A={1,4,9,16,25......}in set builder form
(a) \(\{x : x = n^2 \text{ where } n \in N\}\)
(b) \(\{x : x = n^2 \text{ where } n \in W\}\)
(c) \(\{x : x = n^2 \text{ where } n \in Z\}\)
(d) \(\{x : x = n^2 \text{ where } n \in Q\}\)
Answer: (a) \(\{x : x = n^2 \text{ where } n \in N\}\)
Hint: \(1^2 = 1, 2^2 = 4, 3^2 = 9....\) all are square of natural numbers.

Question. Which of the following is not empty set
(a) A = {x : 1 < x < 2, x is a natural number between 1 and 2}
(b) B = \(\{x : x^2 - 2 = 0 \text{ and x is rational}\}\)
(c) C = {x : x is even prime number > 2}
(d) D = \(\{x : x^2 = 0 \text{ and x is integer}\}\)
Answer: (d) D = \(\{x : x^2 = 0 \text{ and x is integer}\}\)
Hint:
1) \(A = \phi\) because there are no natural numbers between 1 and 2.
2) \(B = \phi\) because \(x^2 = 2 \implies x = \sqrt{2}\) not a rational number
3) \(c = \phi\) because there is only one even prime 2

Question. If A={x/x is a letter in the word "ACCOUNTANCY"} then cardinality of A is
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (c) 7
Hint: Different letters of the word ACCOUNTANCY is {A, C, O, U, N, T, Y} ; Cardinality of A = 7.

Question. Let \(F_1\) be the set of all parallelograms, \(F_2\) be the set of rectangles, \(F_3\) be the set of rhombuses, \(F_4\) be the set of squares and \(F_5\) be the set of trapeziums in a plane then \(F_1 =\)
(a) \(F_2 \cap F_3\)
(b) \(F_2 \cup F_3 \cup F_4\)
(c) \(F_3 \cup F_4 \cup F_5\)
(d) \(F_3 \cap F_1\)
Answer: (b) \(F_2 \cup F_3 \cup F_4\)
Hint: Since every rectangle, rhombus and square is a parallelogram so \(F_1 = F_2 \cup F_3 \cup F_4 \cup F_1\)

Question. If the set of factors of a whole number 'n' including 'n' itself but not '1' is denoted by F(n). If \(F(16) \cap F(40) = F(x)\) then 'x' is
(a) 4
(b) 8
(c) 6
(d) 10
Answer: (b) 8
Hint: \(F(16) = \{2, 4, 8, 16\}\), \(F(40) = \{2, 4, 8, 20, 40\}\)
\(F(16) \cap F(40) = \{2, 4, 8\} = F(8)\)
\(F(x) = F(8) \implies x = 8\)

Question. If A is the set of the divisors of the number 15, B is the set of prime numbers smaller than 10 and C is the set of even numbers smaller than 9, then \((A \cup C) \cap B\) is the set
(a) {1, 3, 5}
(b) {1, 2, 3}
(c) {2, 3, 5}
(d) {2, 5}
Answer: (c) {2, 3, 5}
Hint: \(A = \{1, 3, 5, 15\}\), \(B = \{2, 3, 5, 7\}\), \(C = \{2, 4, 6, 8\}\)
\(\therefore A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 15\}\)
\((A \cup C) \cap B = \{2, 3, 5\}\)

Question. Let A={1,2,3,4,5,6}, B={2,4,6,8} then A - B =
(a) {1,3,5}
(b) {8}
(c) {2,4,6}
(d) \(\phi\)
Answer: (a) {1,3,5}
Hint: A-B= {1,2,3,4,5,6}-{2,4,6,8}={1,3,5}

Question. Let \(A = \{1,2,3,4\}\) and \(B = \{2,3,4,5,6\}\), then \(A \Delta B =\)
(a) {2,3,4}
(b) {1}
(c) {5,6}
(d) {1,5,6}
Answer: (d) {1,5,6}
Hint: \(A \Delta B = \left\{x : x \in A - B \text{ or } x \in B - A\right\}\)

Question. Let U={1,2,3,4,5,6}, A={2,3}, B={3,4,5} then \(A' \cap B' =\) __
(a) {1,2}
(b) {1,6}
(c) {1,5}
(d) {1,4}
Answer: (b) {1,6}
Hint: \(A' = U - A = \{1, 4, 5, 6\}\), \(B' = U - B = \{1, 2, 6\}\)
\(A' \cap B' = \{1, 6\}\)

Question. In a class of 35 students, 24 like to play cricket and 16 like to play football also each student likes to play at least one of the two games. How many students like to play both cricket and football?
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5
Hint: n(C)=24, n(F)=16, n(C \cup F)=35
n(C \cap F) = n(C) + n(F) - n(C \cup F)
= 24 + 16 - 35 = 40 - 35 = 5

Question. In a group of 70 people, 37 like coffee, 52 like tea and each person like atleast one of the two drinks. The number of persons liking both coffee and tea is
(a) 16
(b) 13
(c) 19
(d) 20
Answer: (c) 19
Hint: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
we have, 70 = 37 + 52 - n(A \cap B)

Question. If n(X)=28, n(Y)=32, \(n(X \cup Y)=50\) then \(n(X \cap Y) =\)
(a) 6
(b) 7
(c) 8
(d) 10
Answer: (d) 10
Hint: n(X \cap Y) = n(X) + n(Y) - n(X \cup Y)
= 28 + 32 - 50 = 10

Question. If n(A)=50, n(B)=20 and \(n(A \cap B) = 10\) then \(n(A \Delta B)\) is
(a) 50
(b) 60
(c) 70
(d) 40
Answer: (a) 50
Hint: \(n(A \Delta B) = n(A \cup B) - n(A \cap B)\)
\(= n(A) + n(B) - 2n(A \cap B) = 50\)

Question. The group of beautiful girls is
(a) a null set
(b) A finite set
(c) not a set
(d) Infinite set
Answer: (c) not a set
Hint: Beautiful is a relative term but not well defined.

Question. Which of the following is the roster form of letters of word “SCHOOL”
(a) {s, h, o, l}
(b) {s, c, h, o, l}
(c) {s, c, o, l}
(d) {h, o, o, l}
Answer: (b) {s, c, h, o, l}
Hint: elements not repeated and denoted by small letters.

Question. Write the set {x : x is a positive integer and \(x^2 < 40\)} in the roaster form
(a) {1,2,3,4,5,6}
(b) {1,2,3,4,5,6,7}
(c) {2,3,4,5,6,7}
(d) {0,1,2,3,5,6}
Answer: (a) {1,2,3,4,5,6}
Hint: \(1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36\) all are < 40

Question. Which of the following is finite
(a) A = {x : x is set of points on a line}
(b) B = {x : x \(\in N\) and x is prime}
(c) C = {x : x \(\in N\) and x is odd}
(d) D = {x : x \(\in N\) and (x-1)(x-2)=0}
Answer: (d) D = {x : x \(\in N\) and (x-1)(x-2)=0}
Hint: x-1=0, x-2=0 \(\implies\) x=1,2

Question. Which of the following pairs of sets are equal
(a) A={x : x is letter of word “ALLOY”}, B={x : x is letter of word “LOYAL”}
(b) A={-2,-1,0,1,2}, B={1,2}
(c) A={0}, B={x : x > 5 and x < 15}
(d) A={x : x-5=0}, B={x : \(x^2 = 25\)}
Answer: (a) A={x : x is letter of word “ALLOY”}, B={x : x is letter of word “LOYAL”}
Hint: A={a,l,o,y} B={a,l,o,y}

Question. Let A = {1, {2, 3}} then the number of subsets of A is
(a) 2
(b) 4
(c) 8
(d) 0
Answer: (b) 4
Hint: no of subsets = \(2^n\)

Question. How many elements has P(A), if \(A = \phi\)
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (a) 1
Hint: n(A)=0, \(n[P(A)] = 2^n = 2^0 = 1\)

Question. A={2,4,6,8}, B={6,8,10,12} then \(A \cup B\)
(a) {2,4,6,8,12}
(b) {2,4,6,8,10,12}
(c) {6,8}
(d) {2,4}
Answer: (b) {2,4,6,8,10,12}
Hint: All the elements in A and B.

Question. If A={2,3,4,8,10}, B={3,4,5,10,12}, C={4,5,6,12,14} then \((A \cup B) \cap (A \cup C) =\)
(a) {2,3,4,5,8,10,12}
(b) {2,4,8,10,12}
(c) {3,8,10,12}
(d) {2,8,10}
Answer: (a) {2,3,4,5,8,10,12}
Hint: \(A \cup (B \cap C)\)

Question. Let A={a,e,i,o,u}, B={a,i,k,u} then A-B
(a) {a,e}
(b) {e,i}
(c) {e, o}
(d) {e,i,o}
Answer: (c) {e, o}
Hint: A-B= {a,e,i,o,u} - {a,i,k,u} = {e,o}

Question. Let U={1,2,----,10}, A={1,3,5,7,9} then \(A' =\)
(a) {2,4,6,8,10}
(b) {1,3,5,7,9}
(c) {1,3,2,4}
(d) A
Answer: (a) {2,4,6,8,10}
Hint: \(A' = U - A\)

Question. In a committee 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. The number of persons speaking at least one of these two languages is
(a) 60
(b) 40
(c) 38
(d) 58
Answer: (a) 60
Hint: \(n(S \cup F) = n(S) + n(F) - n(S \cap F)\)
\(\implies n(S \cup F) = 20 + 50 - 10 = 60\)

Question. If A and B are two sets such that \(n(A) = 70\), \(n(B) = 60\) and \(n(A \cup B) = 110\), then \(n(A \cap B)\) is equal to
(a) 240
(b) 50
(c) 40
(d) 20
Answer: (d) 20
Hint: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)

Question. Let \( X = \{1, 2, 3, 4, 5, 6\} \). If \( n \) represent any member of \( X \), then roster form of \( n \in X \) but \( 2n \notin X \) is
(a) \( \{2, 3, 5, 6\} \)
(b) \( \{5, 6\} \)
(c) \( \{4, 5, 6\} \)
(d) \( \{3\} \)
Answer: (c) \( \{4, 5, 6\} \)
Hint: \( A = \{n / n \in X \text{ but } 2n \notin X\} \implies n = 4, 5, 6 \)

Question. Two finite sets have \( m \) and \( n \) elements. If total number of subsets of the first set is 56 more than that of the total number of subsets of the second. The values of \( m \) and \( n \) respectively are
(a) 7,6
(b) 6,3
(c) 5,1
(d) 8, 7
Answer: (b) 6,3
Hint: \( 2^m - 2^n = 56 \). By checking options, if \( m=6, n=3 \), then \( 64 - 8 = 56 \).

Question. If \( A = \{8^n - 7n - 1 : n \in N\} \) and \( B = \{49(n-1) : n \in N\} \) then
(a) \( A \subset B \)
(b) \( B \subseteq A \)
(c) \( A = B \)
(d) \( A \subseteq B \)
Answer: (a) \( A \subset B \)
Hint: \( 8^n = (7+1)^n = {}^n C_0 7^n + {}^n C_1 7^{n-1} + \dots + {}^n C_{n-2} 7^2 + {}^n C_{n-1} 7 + {}^n C_n \). Simplifying \( 8^n - 7n - 1 \), it is a multiple of 49 for all \( n \in N \). \( A \) contains elements which are multiples of 49, while \( B \) contains all multiples of 49. Hence \( A \subset B \).

Question. If \( A = \{\phi, \{\phi\}\} \) then the power set of \( A \) is
(a) \( A \)
(b) \( \{\phi, \{\phi\}, A\} \)
(c) \( \{\phi, \{\phi\}, \{\{\phi\}\}, A\} \)
(d) \( \{\phi\} \)
Answer: (c) \( \{\phi, \{\phi\}, \{\{\phi\}\}, A\} \)
Hint: Number of subsets = \( 2^n \). Since \( n=2 \), total subsets = 4. They are \( \phi, \{\phi\}, \{\{\phi\}\}, A \).

Question. The smallest set \( A \) such that \( A \cup \{1,2\} = \{1,2,3,5,9\} \) is
(a) \( \{2,3,5\} \)
(b) \( \{3,5,9\} \)
(c) \( \{1,2,5,9\} \)
(d) \( \{1,2\} \)
Answer: (b) \( \{3,5,9\} \)
Hint: Since \( A \cup \{1,2\} = \{1,2,3,5,9\} \), set \( A \) must at least contain \( \{3,5,9\} \).

Question. If sets A and B are defined as \( A = \{(x,y) : y = e^x, x \in R\} \) and \( B = \{(x,y) : y = x, x \in R\} \), then
(a) \( B \subset A \)
(b) \( A \subset B \)
(c) \( A \cap B = \phi \)
(d) \( A \cup B = A \)
Answer: (c) \( A \cap B = \phi \)
Hint: The graphs of \( y = e^x \) and \( y = x \) do not intersect.