GSEB Class 11 Maths Solutions Chapter 1 Sets Exercise 1.1

Question 1. Which of the following are sets? Justify your answer?
1. The collection of all months of a year beginning with 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.
Answer:
1. We can clearly identify the members of this collection: January, June, July. So, this collection is well-defined, and therefore, it is a set.
2. The phrase 'most talented' is a vague term. A writer might be considered most talented by one individual but not by another. Consequently, we cannot precisely determine which writers are part of our collection. This collection is not well-defined, so it is not a set.
3. The word 'best' is a vague term. A batsman might be considered the best by one person but not by someone else. So, we cannot definitely judge which batsmen belong to our collection. This collection is not well-defined, which means it is not a set.
4. We can clearly state that the members of this collection are your classmates. This collection is well-defined, making it a set.
5. Clearly, the members of the collection are \(1, 2, 3, \dots, 97, 98, 99\). This collection is well-defined and includes specific numbers, so it is a set.
6. Clearly, the members of the collection are the novels penned by Munshi Prem Chand. This collection is well-defined and precise, so it is a set.
7. We can definitely say that the members of this collection are \(2, 4, 6, \dots\). This collection is well-defined, and thus, it is a set.
8. Clearly, the members of this collection are the various questions found in this chapter. It is well-defined, which means it is a set.
9. The term 'most dangerous' is a vague term. An animal might be considered most dangerous by one individual but not by another. Therefore, it is not a set.
In simple words: A collection is a set if you can clearly tell what belongs in it and what doesn't. If words like 'best' or 'most talented' are used, it's usually not a set because everyone has different ideas.

Exam Tip: To determine if a collection is a set, check if its elements are clearly defined and there is no ambiguity about what belongs to the collection. Vague adjectives usually indicate a non-set.

Question 2. Let \( A = \{1, 2, 3, 4, 5, 6\} \). Insert the appropriate symbol \( \in \) or \( \notin \) in the blank spaces.
1. 5 ______ A
2. 8 ______ A
3. 0 ______ A
4. 4 ______ A
5. 2 ______ A
6. 10 ______ A
Answer:
1. \( 5 \in A \)
2. \( 8 \notin A \)
3. \( 0 \notin A \)
4. \( 4 \in A \)
5. \( 2 \in A \)
6. \( 10 \notin A \)
In simple words: Use \( \in \) if the number is inside the set, and \( \notin \) if the number is not inside the set.

Exam Tip: Remember that \( \in \) means "is an element of" and \( \notin \) means "is not an element of". Carefully check each number against the given set's elements.

Question 3. Write the following sets in the roster form:
1. \( A = \{x : x \text{ is an integer and } - 3 \le x < 7\} \)
2. \( B = \{x: x \text{ is a natural number less than } 6\} \)
3. \( G = \{x : x \text{ is a two digit natural number such that sum of its digits is } 8\} \)
4. \( D = \{x : x \text{ is a prime number which is a divisor of } 60\} \)
5. \( E = \text{the set of all letters in the word TRIGONOMETRY.} \)
6. \( F = \text{the set of all letters in the word BETTER.} \)
Answer:
1. The integers between -3 (inclusive) and 7 (exclusive) are: \( A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\} \)
2. Natural numbers are positive integers, so numbers less than 6 are: \( B = \{1, 2, 3, 4, 5\} \)
3. The two-digit natural numbers whose digits sum to 8 are: \( G = \{17, 26, 35, 44, 53, 62, 71, 80\} \)
4. The prime numbers that divide 60 are 2, 3, and 5: \( D = \{2, 3, 5\} \)
5. The unique letters in the word TRIGONOMETRY are: \( E = \{T, R, I, G, O, N, M, E, Y\} \)
6. The unique letters in the word BETTER are: \( F = \{B, E, T, R\} \)
In simple words: Roster form means listing all the items in the set, separated by commas, inside curly brackets \( \{ \} \). Make sure to list each unique item only once.

Exam Tip: When writing in roster form, ensure you consider all conditions like "integer," "natural number," "prime number," "two-digit," and "unique letters." For natural numbers, remember they start from 1. For integers, include zero and negative numbers if the range allows.

Question 4. Write the following sets in the set builder form:
1. \( \{3, 6, 9, 12\} \)
2. \( \{2, 4, 8, 16, 32\} \)
3. \( \{5, 25, 125, 625\} \)
4. \( \{2, 4, 6, 8, \dots\} \)
5. \( \{1, 4, 9, \dots, 100\} \)
Answer:
1. The numbers are multiples of 3 up to 12. So, \( \{x : x \text{ is a natural number multiple of 3 and } x < 15\} \)
2. The numbers are powers of 2. So, \( \{x : x = 2^n, n \in N \text{ and } n \le 5\} \)
3. The numbers are powers of 5. So, \( \{x : x = 5^n, n \in N \text{ and } n \le 4\} \)
4. The numbers are even natural numbers. So, \( \{x : x \text{ is an even natural number}\} \)
5. The numbers are perfect squares up to 100 (\(10^2\)). So, \( \{x : x = n^2, n \in N \text{ and } n \le 10\} \)
In simple words: Set-builder form means describing the rules or properties that all the items in the set follow, rather than listing them out. You use a variable (like x or n) and then state its properties.

Exam Tip: When converting to set-builder form, identify the pattern (e.g., multiples, powers, squares) and the type of numbers (natural, integer) and their range. Use the notation \( n \in N \) for natural numbers and \( n \in Z \) for integers, as needed.

Question 5. List all the elements of the following sets:
1. \( A = \{x: x \text{ is an odd natural number}\} \)
2. \( B = \{x : x \text{ is an integer, } - \frac{1}{2} < x < \frac{9}{2}\} \)
3. \( C = \{x : x \text{ is an integer, } x^2 \le 4\} \)
4. \( D = \{x: x \text{ is a letter in the word "LOYAL"}\} \)
5. \( E = \{x: x \text{ is a month of a year not having 31 days}\} \)
6. \( F = \{x : x \text{ is a consonant in the English alphabet which precedes k).}\} \)
Answer:
1. Odd natural numbers continue infinitely: \( A = \{1, 3, 5, 7, \dots\} \)
2. Integers greater than \( -0.5 \) and less than \( 4.5 \) are: \( B = \{0, 1, 2, 3, 4\} \)
3. For \( x^2 \le 4 \), integers whose squares are 4 or less are: \( C = \{-2, -1, 0, 1, 2\} \)
4. The unique letters in the word LOYAL are: \( D = \{L, O, Y, A\} \)
5. Months with fewer than 31 days are: \( E = \{\text{February, April, June, September, November}\} \)
6. Consonants that come before the letter 'k' are: \( F = \{b, c, d, f, g, h, j\} \)
In simple words: To list elements, find all the items that fit the rule given for the set. For ranges, be careful about whether the numbers at the ends of the range are included or not.

Exam Tip: Pay close attention to keywords like "natural number," "integer," "odd," "prime," and symbols like \( \le \) or \( < \). For letters in a word, list only the unique letters.

Question 6. Match each of the sets on the left in the roster form with the same set on the right described in the set builder form:
(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) \( \{x: x \text{ is a letter of the word MATHEMATICS}\} \)
Answer:
(i) \( \leftrightarrow \) (c)
(ii) \( \leftrightarrow \) (a)
(iii) \( \leftrightarrow \) (d)
(iv) \( \leftrightarrow \) (b)
In simple words: To match sets, you need to find the description that correctly lists all the elements of a given set. Go through each option and check if it fits.

Exam Tip: For matching questions, it can be helpful to first write out the roster form for each set-builder description, then compare these lists to the given roster form sets.