Differentiation (Part-38) | S N De

Set and Set Representation: 7 MCQ Questions with Answers and Detailed Solutions

In this post, we will practice 7 multiple-choice questions based on Sets and Set Representation. Each question is followed by a detailed explanation of the correct answer.

Topic: Sets and Set Representation
Total Questions: 7
Marks: 1 mark each

Part 1: Questions

Question 1

Which of the following collections does NOT represent a set?

(A) The collection of all students in your class.

(B) The collection of all natural numbers less than 55.

(C) The collection of all 14-month years according to the English calendar.

(D) The collection of all even integers.

Question 2

If \(A=\{1,2,3,4,5,6\}\), which of the following statements is NOT correct?

(A) \(5\in A\)

(B) \(8\in A\)

(C) \(0\notin A\)

(D) \(9\notin A\)

Question 3

If \(A=\{x:x\in\mathbb{N}\text{ and }x<10 following="" is="" of="" p="" set="" the="" then="" which="">

(A) \(\{1,2,3,4,5,6,7,8,9,10\}\)

(B) \(\{1,2,3,4,5,6,7,8,9\}\)

(C) \(\{1,3,5,7,9\}\)

(D) \(\{2,4,6,8,10\}\)

Question 4

If \(B=\{x:x\text{ is a positive integer and }x^2<40 following="" of="" p="" represents="" the="" then="" which="">

(A) \(\{1,3,5,7\}\)

(B) \(\{2,4,6\}\)

(C) \(\{1,2,3,4,5,6\}\)

(D) \(\{1,2,4,5\}\)

Question 5

Let \(C\) be the set of solutions of the equation \[ x^2+x-2=0. \] Which of the following is \(C\)?

(A) \(\{-2,1\}\)

(B) \(\{2,1\}\)

(C) \(\{1,3\}\)

(D) None of these

Question 6

Let \(P\) be the set of all prime numbers that are factors of 48. Which of the following represents \(P\)?

(A) \(\{3,5\}\)

(B) \(\{2,5\}\)

(C) \(\{1,2,3,5\}\)

(D) \(\{2,3\}\)

Question 7

If \(B=\{-1,0,1\}\), which of the following is the correct set-builder form of \(B\)?

(A) \(B=\{x:x\text{ is an integer and }|x|\leq1\}\)

(B) \(B=\{x:|x|\leq1,\ x\in\mathbb{R}\}\)

(C) \(B=\{x:|x|\leq1,\ x\in\mathbb{N}\}\)

(D) None of these

Answer Key

Question	Correct Answer
1	(C)
2	(B)
3	(B)
4	(C)
5	(A)
6	(D)
7	(A)

Part 2: Detailed Solutions

Solution 1

Correct Answer: (C)

A set is a well-defined collection of objects. We must be able to determine clearly whether an object belongs to the collection or not.

Option (A) represents a definite collection because the students in a particular class can be identified.

Option (B) is also a definite collection. The natural numbers less than 55 are

\[ \{1,2,3,\ldots,54\}. \]

Option (D), the collection of all even integers, is also well-defined:

\[ \{\ldots,-4,-2,0,2,4,\ldots\}. \]

Option (C) refers to 14-month years according to the English calendar. The usual English calendar has 12 months, so this collection does not describe the intended well-defined collection in the context of this MCQ.

Therefore, the correct answer is (C).

Solution 2

Correct Answer: (B)

We are given

\[ A=\{1,2,3,4,5,6\}. \]

Therefore, the elements of \(A\) are \(1,2,3,4,5,\) and \(6\).

Since \(5\) is an element of \(A\),

\[ 5\in A. \]

Hence, option (A) is correct.

However, \(8\) is not present in the set. Therefore,

\[ 8\notin A. \]

So the statement \(8\in A\) in option (B) is false.

Similarly,

\[ 0\notin A \] and \[ 9\notin A. \]

Therefore, options (C) and (D) are correct.

Hence, the statement that is NOT correct is (B).

Solution 3

Correct Answer: (B)

We are given

\[ A=\{x:x\in\mathbb{N}\text{ and }x<10 p="">

This means that \(x\) must be a natural number and it must be less than 10.

The natural numbers less than 10 are

\[ 1,2,3,4,5,6,7,8,9. \]

Therefore,

\[ A=\{1,2,3,4,5,6,7,8,9\}. \]

Notice that 10 is not included because the condition is \(x<10 leq10="" not="" p="" x="">

Therefore, the correct answer is (B).

Solution 4

Correct Answer: (C)

We are given

\[ B=\{x:x\text{ is a positive integer and }x^2<40 p="">

We need positive integers whose squares are less than 40.

Check the consecutive positive integers:

\[ 1^2=1<40 p="">

\[ 2^2=4<40 p="">

\[ 3^2=9<40 p="">

\[ 4^2=16<40 p="">

\[ 5^2=25<40 p="">

\[ 6^2=36<40 .="" p="">

But

\[ 7^2=49>40. \]

Therefore, the required positive integers are

\[ 1,2,3,4,5,6. \]

Hence,

\[ B=\{1,2,3,4,5,6\}. \]

Therefore, the correct answer is (C).

Solution 5

Correct Answer: (A)

We have

\[ x^2+x-2=0. \]

Factor the quadratic expression:

\[ x^2+x-2 =x^2+2x-x-2. \]

Taking common factors,

\[ =x(x+2)-1(x+2). \]

Therefore,

\[ =(x-1)(x+2). \]

Hence,

\[ (x-1)(x+2)=0. \]

Therefore,

\[ x-1=0 \] or \[ x+2=0. \]

Thus,

\[ x=1 \] or \[ x=-2. \]

Therefore, the set of solutions is

\[ C=\{-2,1\}. \]

Hence, the correct answer is (A).

Solution 6

Correct Answer: (D)

We need to find the prime factors of 48.

The positive factors of 48 are

\[ 1,2,3,4,6,8,12,16,24,48. \]

A prime number has exactly two positive factors: 1 and the number itself.

Among the factors of 48, the prime numbers are \(2\) and \(3\).

Note that 1 is not a prime number.

Therefore,

\[ P=\{2,3\}. \]

Hence, the correct answer is (D).

Solution 7

Correct Answer: (A)

We are given

\[ B=\{-1,0,1\}. \]

We need to express this set in set-builder form.

Every element of \(B\) is an integer and the absolute value of every element is less than or equal to 1.

In mathematical form,

\[ x\in\mathbb{Z} \] and \[ |x|\leq1. \]

If \(x\) is an integer satisfying \(|x|\leq1\), then the possible values are

\[ x=-1,0,1. \]

Therefore,

\[ B=\{x:x\text{ is an integer and }|x|\leq1\}. \]

Hence, the correct answer is (A).

Important Concepts for Quick Revision

• The symbol \( \in \) means that an element belongs to a set.
• The symbol \( \notin \) means that an element does not belong to a set.
• A set can be represented by listing its elements or by using set-builder notation.
• \(x<10 10.="" 10="" excluded="" includes="" is="" leq10="" li="" means="" that="" whereas="" x="">
• To solve a condition such as \(x^2<40 corresponding="" inequality.="" integers="" li="" or="" positive="" relevant="" test="" the="" use="">
• The solutions of an equation can be written as a set.
• 1 is neither prime nor composite.
• If \(x\in\mathbb{Z}\) and \(|x|\leq1\), then \(x=-1,0,1\).

Final Answers:
1. (C)
2. (B)
3. (B)
4. (C)
5. (A)
6. (D)
7. (A)