Sets: Exercise 1.1 – Guided Solutions

Q1 Which of the following are sets? Justify your answer.

1. The collection of all the months of a year beginning with the letter J.
2. The collection of ten most talented writers of India.
3. A team of eleven best-cricket batsmen of the world.
4. The collection of all boys in your class.
5. The collection of all natural numbers less than 100.
6. A collection of novels written by the writer Munshi Prem Chand.
7. The collection of all even integers.
8. The collection of questions in this Chapter.
9. A collection of most dangerous animals of the world.

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Concept Theory

In mathematics, a set is a well-defined collection of objects. The objects in a set are called elements or members.

A collection is considered a set only if we can clearly determine whether a given object belongs to the collection or not.

If membership is ambiguous or subjective, the collection is not a set.

Example:

• Collection of vowels → well defined → Set
• Collection of beautiful flowers → subjective → Not a set

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Solution Roadmap

To determine whether a given collection is a set, follow these steps:

1. Check whether the objects in the collection are clearly identifiable.
2. If the condition for membership is objective and precise, it forms a set.
3. If the condition depends on opinion, taste, or judgment, it is not a set.

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Solution

(i) The collection of months beginning with the letter J is a set.

The months satisfying the condition are clearly identifiable:

\[ J=\{\text{January},\text{June},\text{July}\} \]

Since the members are well-defined, the collection forms a set.

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(ii) The collection of ten most talented writers of India is not a set.

The word “most talented” is subjective and different people may select different writers. Hence the collection is not well-defined.

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(iii) A team of eleven best cricket batsmen of the world is not a set.

The term “best” does not have a fixed mathematical criterion and depends on personal opinion. Therefore the collection is not well-defined.

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(iv) The collection of all boys in your class is a set.

Membership can be clearly determined because the students in a class are fixed. Hence the collection is well-defined.

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(v) The collection of all natural numbers less than 100 is a set.

\[ N=\{1,2,3,\dots,99\} \]

Each element can be uniquely identified, so the collection is well-defined.

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(vi) The collection of novels written by Munshi Prem Chand is a set.

The novels written by this author are definite and documented, therefore the collection is well-defined.

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(vii) The collection of all even integers is a set.

\[ E=\{2n \mid n \in \mathbb{Z}\} \]

The rule defining the elements is precise, so the collection forms a set.

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(viii) The collection of questions in this chapter is a set.

The chapter contains a fixed number of questions and each question can be identified uniquely.

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(ix) A collection of most dangerous animals of the world is not a set.

The phrase “most dangerous” is vague and depends on interpretation. Thus the collection is not well-defined.

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Significance of This Question

• Introduces the fundamental idea of a well-defined collection.
• Builds the foundation for all later topics in set theory.
• Helps students distinguish between objective and subjective collections.
• Concept frequently appears in JEE / Olympiad conceptual questions.

Q2 Let \(A = \{1,2,3,4,5,6\}\). Insert the appropriate symbol \( \in \) or \( \notin \) in the blanks.

(i) \(5\;...\;A\)
(ii) \(8\;...\;A\)
(iii) \(0\;...\;A\)
(iv) \(4\;...\;A\)
(v) \(2\;...\;A\)
(vi) \(10\;...\;A\)

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Concept Theory

In set theory, the relationship between an element and a set is expressed using the membership symbol.

• \(\in\) → means belongs to the set
• \(\notin\) → means does not belong to the set

For example:

\[ 3 \in \{1,2,3,4\} \]

because 3 is present in the set.

\[ 7 \notin \{1,2,3,4\} \]

because 7 is not included in the set.

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Solution Roadmap

To determine the correct symbol:

1. Look at the elements inside the set \(A\).
2. If the number appears in the set → use \( \in \).
3. If the number is absent → use \( \notin \).

The given set is

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

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Solution

\[ \begin{aligned} (i)\;& 5 \in A \quad &\text{since 5 is an element of the set } A \\ (ii)\;& 8 \notin A \quad &\text{because 8 is not present in } A \\ (iii)\;& 0 \notin A \quad &\text{as 0 does not belong to the set} \\ (iv)\;& 4 \in A \quad &\text{since 4 is included in the set} \\ (v)\;& 2 \in A \quad &\text{because 2 is an element of } A \\ (vi)\;& 10 \notin A \quad &\text{since 10 is not present in } A \end{aligned} \]

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Final Answer

• \(5 \in A\)
• \(8 \notin A\)
• \(0 \notin A\)
• \(4 \in A\)
• \(2 \in A\)
• \(10 \notin A\)

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Significance of This Question

• Introduces the membership concept of sets.
• Helps students understand the symbols \( \in \) and \( \notin \).
• Forms the basis for advanced topics like subset, union, and intersection.
• Commonly appears as concept-check questions in school exams and competitive tests.

Q3 Write the following sets in roster form:

(i) \(A = \{x : x \text{ is an integer and } -3 \le x < 7\}\)
(ii) \(B = \{x : x \text{ is a natural number less than } 6\}\)
(iii) \(C = \{x : x \text{ is a two-digit natural number such that the sum of its digits is } 8\}\)
(iv) \(D = \{x : x \text{ is a prime number which is a divisor of } 60\}\)
(v) \(E =\) The set of all letters in the word TRIGONOMETRY
(vi) \(F =\) The set of all letters in the word BETTER

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Concept Theory

A set can be represented in two common ways:

• Set-builder form: Elements are described using a rule or property.
• Roster form: All elements of the set are listed explicitly inside braces.

For example:

\[ \{x : x \text{ is a natural number less than } 4\} \]

in roster form becomes

\[ \{1,2,3\} \]

While writing sets in roster form:

• List all elements clearly.
• Separate elements with commas.
• Do not repeat elements.

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Solution Roadmap

To convert a set into roster form:

1. Understand the property defining the elements.
2. Determine all elements satisfying that property.
3. Write the elements inside curly braces without repetition.

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Solution

(i) \(A = \{x : x \text{ is an integer and } -3 \le x < 7\}\)

The integers from \(-3\) up to \(6\) satisfy the condition.

\[ A=\{-3,-2,-1,0,1,2,3,4,5,6\} \]

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(ii) \(B = \{x : x \text{ is a natural number less than } 6\}\)

Natural numbers less than 6 are \(1,2,3,4,5\).

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

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(iii) \(C = \{x : x \text{ is a two-digit number with digit sum } 8\}\)

Two-digit numbers whose digits add to 8 are:

\[ 17,26,35,44,53,62,71,80 \] \[ C=\{17,26,35,44,53,62,71,80\} \]

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(iv) \(D = \{x : x \text{ is a prime number which divides } 60\}\)

Prime factors of \(60\) are:

\[ 2,3,5 \] \[ D=\{2,3,5\} \]

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(v) Set of letters in the word TRIGONOMETRY

A set contains only distinct elements, so repeated letters are written once.

\[ E=\{T,R,I,G,O,N,M,E,Y\} \]

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(vi) Set of letters in the word BETTER

Distinct letters in the word BETTER are:

\[ F=\{B,E,T,R\} \]

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Final Answer

\[ \begin{aligned} A &= \{-3,-2,-1,0,1,2,3,4,5,6\} \\ B &= \{1,2,3,4,5\} \\ C &= \{17,26,35,44,53,62,71,80\} \\ D &= \{2,3,5\} \\ E &= \{T,R,I,G,O,N,M,E,Y\} \\ F &= \{B,E,T,R\} \end{aligned} \]

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Significance of This Question

• Teaches conversion from set-builder form to roster form.
• Reinforces understanding of natural numbers, integers, and primes.
• Introduces the rule that sets contain only distinct elements.
• Important foundation for later topics such as subset, union, intersection, and Venn diagrams.

Q4 Write the following sets in the set-builder form :

(i) {3, 6, 9, 12}
(ii) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, . . .}
(v) {1, 4, 9, . . ., 100}

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Concept Theory

A set can be represented in two major forms:

• Roster Form – Elements are listed explicitly.
• Set-Builder Form – Elements are described using a mathematical rule or property.

For example:

\[ \{2,4,6,8\} \]

can be written in set-builder form as

\[ \{x : x = 2n,\; n \in \mathbb{N},\; 1 \le n \le 4\} \]

Here, the rule generating the elements is clearly expressed.

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Solution Roadmap

To convert roster form to set-builder form:

1. Observe the pattern in the numbers.
2. Identify the rule generating the elements.
3. Express the rule using a variable (usually \(n\)).
4. Specify the domain of \(n\).

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Solution

(i) \(A = \{3,6,9,12\}\)

Each element is a multiple of 3.

\[ A = \{x : x = 3n,\; n \in \mathbb{N},\; 1 \le n \le 4\} \]

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(ii) \(B = \{2,4,8,16,32\}\)

These numbers are successive powers of 2.

\[ B = \{x : x = 2^n,\; n \in \mathbb{N},\; 1 \le n \le 5\} \]

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(iii) \(C = \{5,25,125,625\}\)

These elements are powers of 5.

\[ C = \{x : x = 5^n,\; n \in \mathbb{N},\; 1 \le n \le 4\} \]

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(iv) \(D = \{2,4,6,\ldots\}\)

This set represents all even natural numbers.

\[ D = \{x : x = 2n,\; n \in \mathbb{N}\} \]

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(v) \(E = \{1,4,9,\ldots,100\}\)

These are perfect squares from \(1^2\) to \(10^2\).

\[ E = \{x : x = n^2,\; n \in \mathbb{N},\; 1 \le n \le 10\} \]

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Final Answer

\[ \begin{aligned} A &= \{x : x = 3n,\; n \in \mathbb{N},\; 1 \le n \le 4\} \\ B &= \{x : x = 2^n,\; n \in \mathbb{N},\; 1 \le n \le 5\} \\ C &= \{x : x = 5^n,\; n \in \mathbb{N},\; 1 \le n \le 4\} \\ D &= \{x : x = 2n,\; n \in \mathbb{N}\} \\ E &= \{x : x = n^2,\; n \in \mathbb{N},\; 1 \le n \le 10\} \end{aligned} \]

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Significance of This Question

• Builds understanding of pattern recognition in sets.
• Helps students express sets using mathematical rules.
• Forms the basis for topics like subsets, intervals, and sequences.
• Concept is frequently used in JEE / Olympiad algebra problems.

Q5

Q5. List all the elements of the following sets:

(i) \(A = \{x : x \text{ is an odd natural number}\}\)
(ii) \(B = \{x : x \text{ is an integer}, -\frac{1}{2} < x < \frac{9}{2}\}\)
(iii) \(C = \{x : x \text{ is an integer}, x^2 \le 4\}\)
(iv) \(D = \{x : x \text{ is a letter in the word “LOYAL”}\}\)
(v) \(E = \{x : x \text{ is a month of a year not having 31 days}\}\)
(vi) \(F = \{x : x \text{ is a consonant in the English alphabet which precedes k}\}\)

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Concept Theory

A set given in set-builder form describes its elements using a rule or property. To list the elements explicitly, we convert it into roster form.

While listing elements:

• Write all objects satisfying the given condition.
• Avoid repetition of elements.
• Ensure each element clearly satisfies the rule.

Example:

\[ \{x : x \text{ is an even natural number less than } 8\} \] becomes \[ \{2,4,6\} \]

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Solution Roadmap

To list elements of a set:

1. Understand the rule defining the set.
2. Identify all objects satisfying the rule.
3. Write them in roster form inside curly brackets.

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Solution

(i) \(A = \{x : x \text{ is an odd natural number}\}\)

Odd natural numbers are numbers not divisible by 2.

\[ A = \{1,3,5,7,9,\ldots\} \]

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(ii) \(B = \{x : x \text{ is an integer}, -\frac12 < x < \frac92\}\)

The integers between \(-\frac12\) and \(\frac92\) are:

\[ B = \{0,1,2,3,4\} \]

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(iii) \(C = \{x : x \text{ is an integer}, x^2 \le 4\}\)

Since \(x^2 \le 4\), the possible integer values of \(x\) are:

\[ -2,-1,0,1,2 \] \[ C = \{-2,-1,0,1,2\} \]

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(iv) \(D = \{x : x \text{ is a letter in the word “LOYAL”}\}\)

Sets contain only distinct elements.

\[ D = \{L,O,Y,A\} \]

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(v) \(E = \{x : x \text{ is a month not having 31 days}\}\)

Months with fewer than 31 days are:

\[ E = \{\text{February}, \text{April}, \text{June}, \text{September}, \text{November}\} \]

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(vi) \(F = \{x : x \text{ is a consonant preceding } k\}\)

Consonants before the letter \(k\) in the English alphabet are:

\[ F = \{B,C,D,F,G,H,J\} \]

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Final Answer

\[ \begin{aligned} A &= \{1,3,5,7,9,\ldots\} \\ B &= \{0,1,2,3,4\} \\ C &= \{-2,-1,0,1,2\} \\ D &= \{L,O,Y,A\} \\ E &= \{\text{February},\text{April},\text{June},\text{September},\text{November}\} \\ F &= \{B,C,D,F,G,H,J\} \end{aligned} \]

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Significance of This Question

• Strengthens understanding of set-builder to roster conversion.
• Introduces sets involving numbers, words, and real-life objects.
• Reinforces the idea that sets contain only distinct elements.
• Concept forms the basis for advanced topics like subsets and Venn diagrams.

Q6 Match each set written in roster form with the corresponding set written in set-builder form.

\[ \begin{array}{lcl} (i)\; \{1,2,3,6\} & & (a)\; \{x : x \text{ is a prime number and a divisor of }6\}\\ (ii)\; \{2,3\} & & (b)\; \{x : x \text{ is an odd natural number less than }10\}\\ (iii)\; \{M,A,T,H,E,I,C,S\} & & (c)\; \{x : x \text{ is a natural number and divisor of }6\}\\ (iv)\; \{1,3,5,7,9\} & & (d)\; \{x : x \text{ is a letter of the word MATHEMATICS}\} \end{array} \]

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Concept Theory

Sets can be represented either in roster form or in set-builder form.

• Roster form lists all elements explicitly.
• Set-builder form describes elements using a rule or property.

To match the sets, we interpret the rule given in set-builder form and compare it with the listed elements.

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Solution Roadmap

To match the sets correctly:

1. Identify the meaning of the set-builder expression.
2. List the elements satisfying that condition.
3. Compare them with the given roster sets.

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Solution

(i) \( \{1,2,3,6\} \)

These numbers are all natural numbers that divide 6.

Hence it corresponds to

\[ (c)\; \{x : x \text{ is a natural number and divisor of }6\} \]

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

These are the prime numbers that divide 6.

Hence it corresponds to

\[ (a)\; \{x : x \text{ is a prime number and a divisor of }6\} \]

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(iii) \( \{M,A,T,H,E,I,C,S\} \)

These are the distinct letters of the word MATHEMATICS.

Thus it corresponds to

\[ (d)\; \{x : x \text{ is a letter of the word MATHEMATICS}\} \]

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

These are odd natural numbers less than 10.

Hence it corresponds to

\[ (b)\; \{x : x \text{ is an odd natural number less than }10\} \]

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Final Matching

\[ \begin{aligned} (i) &\rightarrow (c) \\ (ii) &\rightarrow (a) \\ (iii) &\rightarrow (d) \\ (iv) &\rightarrow (b) \end{aligned} \]

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Significance of This Question

• Strengthens understanding of roster form and set-builder form.
• Improves ability to interpret mathematical conditions.
• Combines numerical sets with word-based sets.
• Important conceptual foundation for subsets and Venn diagrams.