Set Theory: MCQ Questions 8–23 with Answers and Detailed Solutions
In this post, we will practice 16 multiple-choice questions based on Set Theory, Set Representation, Finite and Infinite Sets, Equal Sets and Equivalent Sets. Each question is followed by a detailed explanation of the correct answer.
Total Questions: 16
Question Numbers: 8–23
Type: Multiple Choice Questions (MCQ)
Part 1: Questions
Question 8
Let \[ C=\{1,3,5,7,\ldots\}. \] Which of the following is the correct set-builder form of \(C\)?
(A) \[ C=\{x:x=2n+1,\;n\in\mathbb{N}\} \]
(B) \[ C=\{x:x=2n-1,\;n\in\mathbb{N}\} \]
(C) \[ C=\{x:x=n-1,\;n\in\mathbb{N}\} \]
(D) None of these
Question 9
Let \[ F=\{1,4,9,16,\ldots,100\}. \] Which of the following is the correct set-builder form of \(F\)?
(A) \[ F=\{x:x=n^2,\;1<n<10,\;n\in\mathbb{N}\} \]
(B) \[ F=\{x:x=n^2,\;1\leq n\leq10,\;n\in\mathbb{R}\} \]
(C) \[ F=\{x:x=n^2,\;1\leq n\leq10,\;n\in\mathbb{N}\} \]
(D) None of these
Question 10
If \[ A=\{a,b,\{c,d\},e\}, \] then which of the following statements is correct?
(A) \(\{c,d\}\subset A\)
(B) \(\{c,d\}\in A\)
(C) \(a\subset A\)
(D) \(\{a,b,e\}\in A\)
Question 11
Let \[ S=\{x:x\text{ is a natural number and }x^2-9=0\}. \] What is the roster form of \(S\)?
(A) \(\{3\}\)
(B) \(\{-3\}\)
(C) \(\{3,-3\}\)
(D) \(\{9,3\}\)
Question 12
Which of the following is the correct roster form of the set \[ \{x:x\text{ is a positive integer and }x^3<70\}? \]
(A) \(\{0,1,2,3,4,5\}\)
(B) \(\{-1,1,2,3,4\}\)
(C) \(\{1,2,3,4\}\)
(D) \(\{0,1,2,3,4\}\)
Question 13
A finite set contains:
(A) Only the empty element
(B) Exactly one element
(C) At least one element
(D) Zero or more elements, but not infinitely many elements
Question 14
Which of the following is a finite set?
(A) The set of all straight lines parallel to a given straight line
(B) The set of all even prime numbers
(C) The set of all odd prime numbers
(D) The set of all concentric circles in a plane
Question 15
Which of the following sets is not equal to the other sets?
(A) \[ A=\{1,2,3\} \]
(B) \[ B=\{x:x^2-2x+1=0\} \]
(C) \[ C=\{x:x^3-6x^2+11x-6=0\} \]
(D) \[ D=\{2,3,1\} \]
Question 16
Given \[ A=\{a,b\},\quad B=\{1,2,3,4\},\quad C=\{6,9,13\}, \] \[ D=\{13,9,6\},\quad E=\{3,0\},\quad F=\{4,2,3,1\}, \] \[ G=\{1,3,5,7\},\quad H=\{0,a\}, \] which of the following statements is correct?
(A) \(A=C\)
(B) \(F=D\)
(C) \(A,E,H\) are not equivalent sets
(D) \(F\) and \(G\) are equivalent sets
Question 17
Which of the following sets are equal?
(A) \[ A=\{5,6\},\quad B=\{x:x\text{ is a root of }x^2-x-30=0\} \]
(B) \[ A=\{n:n\in\mathbb{Z}\text{ and }n^2<36\}, \] \[ B=\{x:x\in\mathbb{R}\text{ and }x^2-3x+2=0\} \]
(C) \[ A=\{x:x\text{ is a perfect square},\;1<x<10\}, \] \[ B=\{x:x\in\mathbb{Z}\text{ and }x^2-13x+36=0\} \]
(D) \[ A=\{x:x\in\mathbb{N},x<5\}, \] \[ B=\{1,2,3\},\quad C=\{1,2,3,2\} \]
Question 18
Which of the following pairs of sets is not equivalent?
(A) \[ A=\{a\},\quad B=\{x:x\text{ is a perfect square}\} \]
(B) \[ A=\{x:x^2-5x+6=0\text{ is satisfied}\}, \quad B=\{2,3\} \]
(C) \[ A=\{1,2,3,4\}, \quad B=\{x:x\text{ is a letter of the word LOYAL}\} \]
(D) \[ A=\{a,b,c,d,e\}, \quad B=\{x:x\in\mathbb{Z}\text{ and }x^2\leq4\} \]
Question 19
Which two sets are equal?
(A) \[ \{0,1,2,3\}\quad\text{and}\quad\{2,3,4,5\} \]
(B) \[ \{-1,3,4,5\}\quad\text{and}\quad\{1,3,5,7\} \]
(C) \[ \{1,3,4,6\}\quad\text{and}\quad\{6,4,3,1\} \]
(D) \[ \{1,2,5\}\quad\text{and}\quad\{2,5,1,4\} \]
Question 20
The set of prime numbers less than \(6\) and the set of prime factors of \(60\) are:
(A) Infinite
(B) Empty
(C) Singleton
(D) Equal
Question 21
If \[ X=\{-1,10,12,16\} \] and \[ Y=\{1,5,12,25\}, \] then which of the following statements is correct about \(X\) and \(Y\)?
(A) \(X\) and \(Y\) are equal sets
(B) \(X\) and \(Y\) are equivalent sets
(C) \(X\) and \(Y\) are empty sets
(D) \(X\) and \(Y\) are infinite sets
Question 22
If \[ A=\{0\}, \] \[ B=\{x:x\text{ is a non-negative root of }x^2+3x=0\}, \] \[ C=\{x:x>10\text{ and }x<5,\;x\text{ is a positive integer}\}, \] and \[ D=\{x:x^2=49,\;x\text{ is a natural number}\}, \] then the correct relation is:
(A) \(A=C\)
(B) \(A=D\)
(C) \(B=C\)
(D) \(A=B\)
Question 23
If \[ A=\{1,2\} \] and \[ B=\{x\in\mathbb{R}:x^2-3x+2=0\}, \] then:
(A) \(A\) and \(B\) are equivalent but not equal sets
(B) \(A\) and \(B\) are equal
(C) \(A\) and \(B\) are not equivalent
(D) \(A\) and \(B\) are infinite sets
Answer Key
| Question | Correct Answer |
|---|---|
| 8 | (B) |
| 9 | (C) |
| 10 | (B) |
| 11 | (A) |
| 12 | (C) |
| 13 | (D) |
| 14 | (B) |
| 15 | (B) |
| 16 | (D) |
| 17 | (D) |
| 18 | (A) |
| 19 | (C) |
| 20 | (D) |
| 21 | (B) |
| 22 | (D) |
| 23 | (B) |
Part 2: Detailed Solutions
Solution 8
Correct Answer: (B)
We are given \[ C=\{1,3,5,7,\ldots\}. \] This is the set of positive odd integers.
For \(n\in\mathbb{N}\), the expression \[ 2n-1 \] generates the numbers \[ 1,3,5,7,9,\ldots \] when \(n=1,2,3,4,\ldots\).
Therefore, \[ C=\{x:x=2n-1,\;n\in\mathbb{N}\}. \]
Option (A), \(2n+1\), starts with \(3\) when \(n=1\), so it misses \(1\). Option (C) gives \(0,1,2,\ldots\), which is not the required set.
Hence, the correct answer is (B).
Solution 9
Correct Answer: (C)
We have \[ F=\{1,4,9,16,\ldots,100\}. \] These are the squares of the natural numbers: \[ 1^2,2^2,3^2,\ldots,10^2. \]
Thus, \[ F=\{x:x=n^2,\;1\leq n\leq10,\;n\in\mathbb{N}\}. \]
Option (A) excludes \(n=1\) and \(n=10\), so it does not include \(1\) and \(100\). Option (B) allows real values of \(n\), which would produce many values that are not members of the given set.
Therefore, the correct answer is (C).
Solution 10
Correct Answer: (B)
Given \[ A=\{a,b,\{c,d\},e\}. \] The elements of \(A\) are: \[ a,\quad b,\quad \{c,d\},\quad e. \]
Notice carefully that \(\{c,d\}\) itself is an element of \(A\). Therefore, \[ \{c,d\}\in A. \]
However, \(c\) and \(d\) individually are not elements of \(A\). Hence \[ \{c,d\}\not\subset A. \]
Also, \(a\) is an element, not a set, so \(a\subset A\) is not correct.
Therefore, the correct answer is (B).
Solution 11
Correct Answer: (A)
We have \[ x^2-9=0. \] Using the difference of two squares, \[ x^2-3^2=0. \] Therefore, \[ (x-3)(x+3)=0. \]
Hence, \[ x=3\quad\text{or}\quad x=-3. \]
But \(x\) is specified to be a natural number. Therefore, \(-3\) is not allowed. Thus, \[ S=\{3\}. \]
Hence, the correct answer is (A).
Solution 12
Correct Answer: (C)
We need positive integers satisfying \[ x^3<70. \]
Check the positive integers:
\[ 1^3=1<70 \] \[ 2^3=8<70 \] \[ 3^3=27<70 \] \[ 4^3=64<70 \] But \[ 5^3=125>70. \]Therefore, the required positive integers are \[ 1,2,3,4. \]
Hence the set is \[ \{1,2,3,4\}. \]
Therefore, the correct answer is (C).
Solution 13
Correct Answer: (D)
A finite set is a set having a limited number of elements. The number of elements may be zero, one, two, three, or any other finite number.
For example, \[ \varnothing \] is a finite set containing zero elements, while \[ \{1,2,3\} \] is a finite set containing three elements.
Therefore, a finite set may contain zero or more elements, but it cannot contain infinitely many elements.
Hence, the correct answer is (D).
Solution 14
Correct Answer: (B)
We examine the options.
(A) There are infinitely many lines parallel to a given line. Therefore, it is an infinite set.
(B) The only even prime number is \[ 2. \] Therefore, the set of all even prime numbers is \[ \{2\}, \] which is a finite set.
(C) There are infinitely many odd prime numbers. Therefore, this is an infinite set.
(D) There are infinitely many concentric circles with the same centre and different radii. Therefore, this is an infinite set.
Thus, the correct answer is (B).
Solution 15
Correct Answer: (B)
We compare the sets.
For (A), \[ A=\{1,2,3\}. \]
For (B), \[ x^2-2x+1=0. \] Factorising, \[ (x-1)^2=0. \] Therefore, \[ x=1. \] Hence, \[ B=\{1\}. \]
For (C), \[ x^3-6x^2+11x-6=0. \] Factorising, \[ (x-1)(x-2)(x-3)=0. \] Therefore, \[ C=\{1,2,3\}. \]
Finally, \[ D=\{2,3,1\}=\{1,2,3\}. \] The order of elements does not matter in a set.
Thus, \[ A=C=D=\{1,2,3\}, \] whereas \[ B=\{1\}. \]
Therefore, (B) is not equal to the other sets.
Solution 16
Correct Answer: (D)
Two sets are called equivalent if they have the same number of elements, even if their elements are different.
We have \[ F=\{4,2,3,1\} \] and \[ G=\{1,3,5,7\}. \]
Both sets contain four distinct elements: \[ n(F)=4,\qquad n(G)=4. \] Therefore, \[ n(F)=n(G). \]
Hence \(F\) and \(G\) are equivalent sets. They are not equal sets because their elements are different.
Therefore, the correct answer is (D).
Solution 17
Correct Answer: (D)
Consider option (D): \[ A=\{x:x\in\mathbb{N},x<5\}. \] Therefore, \[ A=\{1,2,3,4\}. \]
We are also given \[ B=\{1,2,3\} \] and \[ C=\{1,2,3,2\}. \]
In a set, repeated elements are written only once. Therefore, \[ C=\{1,2,3\}. \] Thus, \[ B=C. \]
Therefore, option (D) correctly identifies equal sets.
For example, in option (A), \[ x^2-x-30=0 \] gives \[ (x-6)(x+5)=0, \] so the roots are \(6\) and \(-5\), not \(\{5,6\}\).
Hence, the correct answer is (D).
Solution 18
Correct Answer: (A)
Equivalent sets must have the same number of elements.
In (A), \[ A=\{a\} \] contains only one element.
But \[ B=\{x:x\text{ is a perfect square}\} \] contains infinitely many elements: \[ 1,4,9,16,25,\ldots \] Therefore, \(A\) and \(B\) are not equivalent.
For (B), \[ x^2-5x+6=0 \] gives \[ (x-2)(x-3)=0, \] so the solution set is \[ \{2,3\}. \] Thus it is equivalent to \(B=\{2,3\}\).
For (C), the distinct letters of LOYAL are \[ \{L,O,Y,A\}, \] which has four elements, the same as \(\{1,2,3,4\}\).
For (D), \[ x^2\leq4,\quad x\in\mathbb{Z} \] gives \[ x=-2,-1,0,1,2. \] Thus there are five elements, the same as \(A=\{a,b,c,d,e\}\).
Therefore, the correct answer is (A).
Solution 19
Correct Answer: (C)
Two sets are equal when they contain exactly the same elements, regardless of their order.
Consider option (C): \[ \{1,3,4,6\} \] and \[ \{6,4,3,1\}. \]
Both contain exactly the same elements: \[ 1,3,4,6. \] Therefore, \[ \{1,3,4,6\}=\{6,4,3,1\}. \]
The order in which elements are written does not affect a set.
Hence, the correct answer is (C).
Solution 20
Correct Answer: (D)
The prime numbers less than \(6\) are \[ 2,3,5. \] Therefore, the first set is \[ \{2,3,5\}. \]
Now factorise \(60\): \[ 60=2^2\times3\times5. \] Therefore, the distinct prime factors of \(60\) are \[ \{2,3,5\}. \]
Hence, both sets are exactly the same: \[ \{2,3,5\}=\{2,3,5\}. \]
Therefore, the correct answer is (D), Equal.
Solution 21
Correct Answer: (B)
We are given \[ X=\{-1,10,12,16\} \] and \[ Y=\{1,5,12,25\}. \]
The elements of \(X\) and \(Y\) are not the same. Therefore, \[ X\neq Y. \]
However, each set contains four elements: \[ n(X)=4 \] and \[ n(Y)=4. \]
Therefore, \[ n(X)=n(Y), \] so \(X\) and \(Y\) are equivalent sets.
Hence, the correct answer is (B).
Solution 22
Correct Answer: (D)
We have \[ A=\{0\}. \]
For \(B\), \[ x^2+3x=0. \] Factorising, \[ x(x+3)=0. \] Therefore, \[ x=0\quad\text{or}\quad x=-3. \]
The non-negative root is \(0\). Hence, \[ B=\{0\}. \]
Therefore, \[ A=B. \]
For \(C\), we need a positive integer satisfying \[ x>10\quad\text{and}\quad x<5. \] No number can satisfy both conditions simultaneously. Hence, \[ C=\varnothing. \]
For \(D\), \[ x^2=49. \] Thus, \[ x=\pm7. \] Since \(x\) is a natural number, \[ D=\{7\}. \]
Therefore, \[ A=B=\{0\}. \]
Hence, the correct answer is (D).
Solution 23
Correct Answer: (B)
We are given \[ A=\{1,2\}. \]
For \(B\), \[ x^2-3x+2=0. \] Factorising, \[ (x-1)(x-2)=0. \] Therefore, \[ x=1\quad\text{or}\quad x=2. \]
Hence, \[ B=\{1,2\}. \]
Since \[ A=\{1,2\} \] and \[ B=\{1,2\}, \] we get \[ A=B. \]
Thus the two sets are not merely equivalent; they are exactly equal.
Therefore, the correct answer is (B).
Important Concepts for Quick Revision
- Set-builder form: A set can be represented by describing the common property of its elements.
- Roster form: The elements of a set are listed inside braces. For example, \[ \{1,2,3\}. \]
- Element symbol: \(x\in A\) means \(x\) is an element of \(A\).
- Not an element: \(x\notin A\) means \(x\) is not an element of \(A\).
- Equal sets: Two sets are equal if they contain exactly the same elements.
- Order does not matter: \[ \{1,2,3\}=\{3,2,1\}. \]
- Repeated elements are ignored: \[ \{1,2,3,2\}=\{1,2,3\}. \]
- Equivalent sets: Two sets are equivalent if they have the same number of elements.
- Finite set: A finite set contains a limited number of elements. The empty set is also finite.
- Infinite set: A set containing infinitely many elements is called an infinite set.
- Prime number: \(1\) is neither prime nor composite. The only even prime number is \(2\).
- For a quadratic equation, the roots can be used to form a set. For example, \[ x^2-3x+2=0 \] has roots \(1\) and \(2\), so its solution set is \[ \{1,2\}. \]
- When a set is defined using a condition such as \[ x^2<40, \] check the values satisfying the condition carefully.
Final Answers
8. (B)
9. (C)
10. (B)
11. (A)
12. (C)
13. (D)
14. (B)
15. (B)
16. (D)
17. (D)
18. (A)
19. (C)
20. (D)
21. (B)
22. (D)
23. (B)
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