JEE Mathematics Sets MCQs Set 04

Mathematics Objective Questions and Answers: Sets

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Access the complete set of multiple-choice questions for Sets below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.

Multiple Choice Questions

Question 1: If \( U=\{1,2,\dots,10\} \), \( A=\{1,2,3,4\} \) and \( B=\{2,4,6,8,10\} \), then \( (A\cup B)' \) equals
(a) \( \{5,7,9\} \)
(b) \( \{1,3,5,7\} \)
(c) \( \{1,3,5,6,7,8\} \)
(d) \( \{2,4\} \)
Show Answer & Explanation

Answer: (a) \( \{5,7,9\} \)

Explanation:
1. \( A\cup B=\{1,2,3,4,6,8,10\} \).
2. Removing these from \( U \) leaves \( \{5,7,9\} \).

Teacher's Note:
1. Take the union first, then the complement in U.

Question 2: If \( A=\{1,2,3,4\} \), \( B=\{2,3,5,6\} \) and \( C=\{3,4,6,7\} \), then \( A-(B\cap C) \) equals
(a) \( \{1,3,4\} \)
(b) \( \{1,2,4\} \)
(c) \( \{2,3\} \)
(d) \( \{1,2,3,4\} \)
Show Answer & Explanation

Answer: (b) \( \{1,2,4\} \)

Explanation:
1. \( B\cap C=\{3,6\} \).
2. Removing 3 (6 is not in A) from A gives \( \{1,2,4\} \).

Teacher's Note:
1. Intersect first, then subtract.

Question 3: Let \( V=\{a,e,i,o,u\} \) and \( B=\{a,i,k,u\} \). Then \( V-B \) and \( B-V \) are respectively
(a) \( \{e,o\} \) and \( \{k\} \)
(b) \( \{e\} \) and \( \{k\} \)
(c) \( \{o\} \) and \( \{k\} \)
(d) \( \{e,o\} \) and \( \{k,i\} \)
Show Answer & Explanation

Answer: (a) \( \{e,o\} \) and \( \{k\} \)

Explanation:
1. \( V-B \) keeps the vowels not in B: e and o.
2. \( B-V \) keeps the elements of B not in V: only k.

Teacher's Note:
1. A - B contains elements of A that are not in B.

Question 4: If A and B are finite sets, which one of the following is correct?
(a) \( n(A-B)=n(A)-n(B) \)
(b) \( n(A-B)=n(A)-n(A\cap B) \)
(c) \( n(A-B)=n(B)-n(A\cap B) \)
(d) \( n(A-B)=n(B-A) \)
Show Answer & Explanation

Answer: (b) \( n(A-B)=n(A)-n(A\cap B) \)

Explanation:
1. A splits into the disjoint parts \( A-B \) and \( A\cap B \).
2. So \( n(A)=n(A-B)+n(A\cap B) \), i.e. \( n(A-B)=n(A)-n(A\cap B) \).

Teacher's Note:
1. Split A into the part inside B and the part outside B.

Question 5: In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea. If every person drinks at least one of tea and coffee, how many drink coffee but not tea?
(a) 17
(b) 36
(c) 23
(d) 19
Show Answer & Explanation

Answer: (d) 19

Explanation:
1. The number who drink tea is 33, so those who drink coffee but not tea = 52 - 33 = 19.
2. Check: 17 drink both (33 - 16), 16 only tea and 19 only coffee, total 52.

Teacher's Note:
1. Persons not drinking tea must drink only coffee.

Question 6: In a school of 600 students, 400 can speak Telugu and 300 can speak Hindi. If every student speaks at least one of the two languages, the number who can speak both is
(a) 100
(b) 200
(c) 400
(d) 300
Show Answer & Explanation

Answer: (a) 100

Explanation:
1. \( n(T\cap H)=n(T)+n(H)-n(T\cup H)=400+300-600=100 \).

Teacher's Note:
1. Use \( n(A\cup B)=n(A)+n(B)-n(A\cap B) \).

Question 7: In a group of 500 students, 475 can speak Hindi and 200 can speak Bengali. If everyone speaks at least one of these, the number of students who speak Hindi only is
(a) 275
(b) 300
(c) 325
(d) 350
Show Answer & Explanation

Answer: (b) 300

Explanation:
1. Both: \( 475+200-500=175 \).
2. Hindi only: \( 475-175=300 \) (equivalently 500 - 200).

Teacher's Note:
1. Hindi only = total - those who speak Bengali.

Question 8: A survey shows that 63% of the people watch one news channel and 76% watch another. If x% watch both channels (everyone watches at least one), then
(a) \( x=35 \)
(b) \( x=63 \)
(c) \( 39\le x\le 63 \)
(d) \( x=39 \)
Show Answer & Explanation

Answer: (c) \( 39\le x\le 63 \)

Explanation:
1. \( x=63+76-n(\text{union})=139-n \) where \( n\le 100 \), so \( x\ge 39 \).
2. Also \( x\le 63 \), as the intersection cannot exceed the smaller set. Hence \( 39\le x\le 63 \).

Teacher's Note:
1. The union cannot exceed 100%.

Question 9: If \( A=\{x:x=n^2,\ n=1,2,3\} \), the number of proper subsets of A is
(a) 3
(b) 8
(c) 7
(d) 4
Show Answer & Explanation

Answer: (c) 7

Explanation:
1. \( A=\{1,4,9\} \) has 3 elements, so \( 2^3=8 \) subsets.
2. Proper subsets exclude A itself: 7.

Teacher's Note:
1. Proper subsets = \( 2^n-1 \).

Question 10: The number of elements in \( P[P(P(\phi))] \) is
(a) 2
(b) 3
(c) 4
(d) 5
Show Answer & Explanation

Answer: (c) 4

Explanation:
1. \( P(\phi)=\{\phi\} \) has 1 element.
2. \( P(P(\phi)) \) has \( 2^1=2 \) elements.
3. \( P[P(P(\phi))] \) has \( 2^2=4 \) elements.

Teacher's Note:
1. Apply \( n(P(X))=2^{n(X)} \) repeatedly.

Question 11: In a battle 70% of the combatants lost one eye, 80% an ear, 75% an arm and 85% a leg. The minimum percentage who lost all four is
(a) 10
(b) 12
(c) 15
(d) none of these
Show Answer & Explanation

Answer: (a) 10

Explanation:
1. The percentages not losing each are 30, 20, 25 and 15, which total 90.
2. At most 90% can have escaped at least one loss, so at least \( 100-90=10\% \) lost all four.

Teacher's Note:
1. Add the complements and subtract from 100.

Question 12: If \( A=\{x:x \) is a multiple of \( 4\} \) and \( B=\{x:x \) is a multiple of \( 6\} \), then \( A\cap B \) consists of all multiples of
(a) 16
(b) 12
(c) 8
(d) 4
Show Answer & Explanation

Answer: (b) 12

Explanation:
1. A common multiple of 4 and 6 is a multiple of their LCM.
2. LCM(4, 6) = 12.

Teacher's Note:
1. The intersection is the set of common multiples.

Question 13: If \( A=\{1,2,4\} \), \( B=\{2,4,5\} \) and \( C=\{2,5\} \), then \( (A-B)\times(B-C) \) is
(a) \( \{(1,2),(1,5),(2,5)\} \)
(b) \( \{(1,4)\} \)
(c) \( (1,4) \)
(d) none of these
Show Answer & Explanation

Answer: (b) \( \{(1,4)\} \)

Explanation:
1. \( A-B=\{1\} \) and \( B-C=\{4\} \).
2. Their Cartesian product is \( \{(1,4)\} \).

Teacher's Note:
1. The Cartesian product is a set of ordered pairs.

Question 14: If A and B are any two sets, \( A\cap(A\cap B)' \) equals
(a) A
(b) B
(c) \( \phi \)
(d) \( A\cap B^c \)
Show Answer & Explanation

Answer: (d) \( A\cap B^c \)

Explanation:
1. \( A\cap(A\cap B)'=A\cap(A'\cup B') \) by De Morgan.
2. \( =(A\cap A')\cup(A\cap B')=\phi\cup(A\cap B')=A\cap B^c \).

Teacher's Note:
1. Apply De Morgan's law, then distribute.

Question 15: 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) none of these
Show Answer & Explanation

Answer: (b) \( \{3,5,9\} \)

Explanation:
1. A must contain 3, 5 and 9, which are not in \( \{1,2\} \).
2. 1 and 2 are already supplied, so the smallest A is \( \{3,5,9\} \).

Teacher's Note:
1. A must supply exactly the missing elements.

Question 16: The set \( A=\{x\in\mathbb R:x^2=16\text{ and }2x=6\} \) equals
(a) \( \phi \)
(b) \( \{14,3,4\} \)
(c) \( \{3\} \)
(d) \( \{4\} \)
Show Answer & Explanation

Answer: (a) \( \phi \)

Explanation:
1. \( x^2=16 \) gives \( x=\pm4 \), and \( 2x=6 \) gives \( x=3 \).
2. No real number satisfies both, so \( A=\phi \).

Teacher's Note:
1. Both conditions must hold at once.

Question 17: The set builder form of \( A=\{3,6,9,12\} \) and \( B=\{1,4,9,\dots,100\} \) is
(a) \( A=\{x:x=3n,n\in N,1\le n\le5\} \), \( B=\{x:x=n^2,n\in N,1\le n\le10\} \)
(b) \( A=\{x:x=3n,n\in N,1\le n\le4\} \), \( B=\{x:x=n^2,n\in N,1\le n\le10\} \)
(c) \( A=\{x:x=3n,n\in N,1\le n\le4\} \), \( B=\{x:x=n^2,n\in N,1(d) none of these
Show Answer & Explanation

Answer: (b) \( A=\{x:x=3n,n\in N,1\le n\le4\} \), \( B=\{x:x=n^2,n\in N,1\le n\le10\} \)

Explanation:
1. \( A \) has 4 terms \( 3n \) with \( n=1 \) to 4.
2. \( B \) has the squares \( 1^2,\dots,10^2 \), so \( 1\le n\le10 \).

Teacher's Note:
1. Check the first and last term of each set.

Question 18: Which of the following sets is finite?
(a) \( \{x\in Z:x^2-5x+6=0\} \)
(b) \( \{x\in Z:x^2\text{ is even}\} \)
(c) \( \{x\in Z:x>-10\} \)
(d) all of these
Show Answer & Explanation

Answer: (a) \( \{x\in Z:x^2-5x+6=0\} \)

Explanation:
1. The first set is \( \{2,3\} \), which is finite.
2. Squares that are even occur for all even integers, and \( x>-10 \) includes all larger integers, so those are infinite.

Teacher's Note:
1. A finite set can be listed completely.

Question 19: Which of the following is a singleton set?
(a) \( \{x:|x|=5,x\in N\} \)
(b) \( \{x:|x|=6,x\in Z\} \)
(c) \( \{x:x^2+2x+1=0,x\in N\} \)
(d) \( \{x:x^2=7,x\in N\} \)
Show Answer & Explanation

Answer: (a) \( \{x:|x|=5,x\in N\} \)

Explanation:
1. (a) gives \( \{5\} \), one element.
2. (b) gives \( \{-6,6\} \); (c) has root \( -1\notin N \) and (d) has no natural root.

Teacher's Note:
1. Solve each condition inside the stated universe.

Question 20: Which of the following is not a null set?
(a) The set of odd natural numbers divisible by 2
(b) The set of even prime numbers
(c) \( \{x:x \) is a natural number, \( x<5 \) and \( x>7\} \)
(d) \( \{y:y \) is a point common to any two parallel lines\( \} \)
Show Answer & Explanation

Answer: (b) The set of even prime numbers

Explanation:
1. Odd numbers are never divisible by 2, no natural number is below 5 and above 7, and parallel lines share no point: (a), (c), (d) are empty.
2. The only even prime is 2, so (b) is \( \{2\} \).

Teacher's Note:
1. Check whether each set has at least one element.

Question 21: Let \( U=\{1,2,\dots,10\} \), \( A=\{1,2,3,4,5\} \), \( B=\{2,4,6,7\} \) and \( C=\{2,3,4,8\} \). Which of the following is true?
(a) \( (B\cup C)'=\{1,5,9,10\} \)
(b) \( (C-A)'=\{1,2,3,5,6,7,8,9,10\} \)
(c) both (a) and (b)
(d) none of the above
Show Answer & Explanation

Answer: (a) \( (B\cup C)'=\{1,5,9,10\} \)

Explanation:
1. \( B\cup C=\{2,3,4,6,7,8\} \), so its complement is \( \{1,5,9,10\} \): (a) is true.
2. \( C-A=\{8\} \), whose complement is \( \{1,2,3,4,5,6,7,9,10\} \), so (b) is false.

Teacher's Note:
1. Evaluate each statement separately.

Question 22: If \( A\subset B \) and \( B\subset C \), then which of the following is true?
(a) \( B-A=C-B \)
(b) \( A\cap B\cap C=B \)
(c) \( A\cup B=B\cap C \)
(d) \( A\cup B\cup C=A \)
Show Answer & Explanation

Answer: (c) \( A\cup B=B\cap C \)

Explanation:
1. \( A\subset B\subset C \) gives \( A\cup B=B \) and \( B\cap C=B \), so (c) holds.
2. \( A\cap B\cap C=A \) and \( A\cup B\cup C=C \), so (b) and (d) fail.

Teacher's Note:
1. Put the sets in a chain A inside B inside C.

Question 23: Consider \( A=\{1,2,3\} \), \( B=\{x\in R:x^2-2x+1=0\} \), \( C=\{1,2,2,3\} \) and \( D=\{x\in R:x^3-6x^2+11x-6=0\} \). Which of these are equal?
(a) \( A=B=C \)
(b) \( A=C=D \)
(c) \( A=B=D \)
(d) \( B=C=D \)
Show Answer & Explanation

Answer: (b) \( A=C=D \)

Explanation:
1. B = {1}. C = {1, 2, 3}, since repeated elements are counted once.
2. \( x^3-6x^2+11x-6=(x-1)(x-2)(x-3) \) so D = {1, 2, 3}. Hence A = C = D.

Teacher's Note:
1. List every set explicitly.

Question 24: Consider \( A=\{0\} \), \( B=\{x:x>15\text{ and }x<5\} \), \( C=\{x:x-5=0\} \), \( D=\{x:x^2=25\} \) and \( E=\{x:x \) is a positive integral root of \( x^2-2x-15=0\} \). Choose the pair of equal sets.
(a) A and B
(b) C and D
(c) C and E
(d) B and C
Show Answer & Explanation

Answer: (c) C and E

Explanation:
1. B is empty, C = {5}, D = {-5, 5}.
2. \( x^2-2x-15=(x-5)(x+3) \); the positive root is 5, so E = {5} = C.

Teacher's Note:
1. List all five sets.

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