Exercise 1.1 – ncerthelp.in

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Exercise 1.1

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Class 11 Maths Chapter 1: Sets – Complete Notes & NCERT Solutions

Exercise 1.1 - Sets - Complete Solutions

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

(i) The collection of all the months of a year beginning with the letter J.

(ii) The collection of ten most talented writers of India.

(iii) A team of eleven best-cricket batsmen of the world.

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

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

(vi) A collection of novels written by the writer Munshi Prem Chand.

(vii) The collection of all even integers.

(viii) The collection of questions in this chapter.

(ix) A collection of most dangerous animals of the world.

💡 Click to View Solution

1 Understanding "Well-Defined Collection"

A set is a collection where we can clearly say YES or NO for any object. If there's ambiguity or different opinions, it's NOT a set.

2 Solution for each part

(i) Months beginning with J:
✓ YES - This IS a SET
Reason: The collection consists of exactly: January, June, July. It is well-defined and everyone agrees on these months.

(ii) Ten most talented writers of India:
✗ NO - NOT a SET
Reason: A writer may be "most talented" for one person but not for another person. Opinion varies from person to person. So, the given collection is not well-defined.

(iii) Eleven best-cricket batsmen of the world:
✗ NO - NOT a SET
Reason: The term "best cricket batsman" is vague. The same batsman may be considered one of the best for one person but not for another. Opinion varies. Not well-defined.

(iv) All boys in your class:
✓ YES - This IS a SET
Reason: Any boy is either in your class or not in your class. There is no ambiguity. The given collection is well-defined.

(v) All natural numbers less than 100:
✓ YES - This IS a SET
Reason: The collection consists of first 99 natural numbers {1, 2, 3, ..., 99}. It is well-defined and clear.

(vi) Novels written by Munshi Prem Chand:
✓ YES - This IS a SET
Reason: It is a well-defined collection. We can find a list of all his novels.

(vii) All even integers:
✓ YES - This IS a SET
Reason: It is a well-defined collection {..., -4, -2, 0, 2, 4, ...}

(viii) Questions in this chapter:
✓ YES - This IS a SET
Reason: It is a well-defined collection.

(ix) Most dangerous animals of the world:
✗ NO - NOT a SET
Reason: The criterion for "most dangerous" varies from person to person. For some people, even a lizard is very dangerous. Not well-defined.

✓ SETS: (i), (iv), (v), (vi), (vii), (viii)
✗ NOT SETS: (ii), (iii), (ix)

2 Insert the appropriate symbol ∈ or ∉ in the blank spaces

Let A = {1, 2, 3, 4, 5, 6}

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

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1 Understanding the Symbols

∈ means "is an element of" (the number IS in the set)
∉ means "is not an element of" (the number is NOT in the set)

A = {1, 2, 3, 4, 5, 6} contains exactly these 6 numbers.

2 Check each number

(i) 5 ... A → Is 5 in set A? YES! ∴ 5 ∈ A

(ii) 8 ... A → Is 8 in set A? NO! ∴ 8 ∉ A

(iii) 0 ... A → Is 0 in set A? NO! ∴ 0 ∉ A

(iv) 4 ... A → Is 4 in set A? YES! ∴ 4 ∈ A

(v) 2 ... A → Is 2 in set A? YES! ∴ 2 ∈ A

(vi) 10 ... A → Is 10 in set A? NO! ∴ 10 ∉ A

(i) 5 ∈ A (ii) 8 ∉ A (iii) 0 ∉ A
(iv) 4 ∈ A (v) 2 ∈ A (vi) 10 ∉ A

3 Write the following sets in roster form

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

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

(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {x : x is a prime number which is 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

💡 Click to View Solution

1 What is Roster Form?

Roster form means listing all elements inside curly brackets {...}
Instead of describing the set, we write down every element.

2 Solution for each part

(i) A = {x : x is an integer and -3 ≤ x < 7}
Integers from -3 to 6 (not including 7):
A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {x : x is a natural mber less than 6}
Natural numbers are 1, 2, 3, 4, 5, ... (less than 6 means up to 5)
B = {1, 2, 3, 4, 5}

(iii) C = {x : x is a two-digit number with digit sum = 8}
Find all two-digit numbers: 17(1+7=8), 26(2+6=8), 35(3+5=8), 44(4+4=8), 53(5+3=8), 62(6+2=8), 71(7+1=8), 80(8+0=8)
C = {17, 26, 35, 44, 53, 62, 71, 80}

(iv) D = {x : x is a prime number which is divisor of 60}
Divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Prime divisors: 2, 3, 5
D = {2, 3, 5}

(v) E = Letters in TRIGONOMETRY
Letters: T, R, I, G, O, N, O, M, E, T, R, Y
Drop repetitions (T, R, O repeat):
E = {T, R, I, G, O, N, M, E, Y}

(vi) F = Letters in BETTER
Letters: B, E, T, T, E, R
Drop repetitions (E and T repeat):
F = {B, E, T, R}

4 Write the following sets in 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}

💡 Click to View Solution

1 What is Set-Builder Form?

Set-builder form describes the set using a rule.
Format: {x : property of x} or {x : condition}
Instead of listing, we describe using a pattern/rule.

2 Solution for each part

(i) {3, 6, 9, 12}
Pattern: 3×1, 3×2, 3×3, 3×4 (multiples of 3)
{x : x = 3n, n ∈ ℕ and 1 ≤ n ≤ 4}

(ii) {2, 4, 8, 16, 32}
Pattern: 2¹, 2², 2³, 2⁴, 2⁵ (powers of 2)
{x : x = 2ⁿ, n ∈ ℕ and 1 ≤ n ≤ 5}

(iii) {5, 25, 125, 625}
Pattern: 5¹, 5², 5³, 5⁴ (powers of 5)
{x : x = 5ⁿ, n ∈ ℕ and 1 ≤ n ≤ 4}

(iv) {2, 4, 6, ...}
Pattern: 2×1, 2×2, 2×3, ... (even natural numbers)
{x : x = 2n, n ∈ ℕ}
Alternatively: {x : x is an even natural number}

(v) {1, 4, 9, ..., 100}
Pattern: 1², 2², 3², ..., 10² (perfect squares up to 100)
{x : x = n², n ∈ ℕ and 1 ≤ n ≤ 10}

5 List all the elements of the following sets

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

(ii) B = {x : x is an integer, -1/2 < x < 9/2}

(iii) C = {x : x is an integer, x² ≤ 4}

(iv) D = {x : x is a letter in the word "LOYAL"}

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

(vi) F = {x : x is a consonant in the English alphabet which precedes k}

💡 Click to View Solution

1 Understanding the Task

We need to find what elements satisfy the given condition and list them all.

2 Solution for each part

(i) A = {x : x is an odd natural number}
Odd natural numbers: 1, 3, 5, 7, 9, 11, ...
A = {1, 3, 5, ...} (infinite set)

(ii) B = {x : x is an integer, -1/2 < x < 9/2}
-1/2 = -0.5 and 9/2 = 4.5
Integers between -0.5 and 4.5: 0, 1, 2, 3, 4
B = {0, 1, 2, 3, 4}

(iii) C = {x : x is an integer, x² ≤ 4}
Which integers have x² ≤ 4?
(-2)² = 4 ✓, (-1)² = 1 ✓, 0² = 0 ✓, 1² = 1 ✓, 2² = 4 ✓
C = {-2, -1, 0, 1, 2}

(iv) D = {x : x is a letter in the word "LOYAL"}
Letters in LOYAL: L, O, Y, A, L (drop repetition of L)
D = {L, O, Y, A}

(v) E = {x : x is a month not having 31 days}
Months with 31 days: Jan, Mar, May, Jul, Aug, Oct, Dec
Months without 31 days: Feb (28/29), Apr (30), Jun (30), Sep (30), Nov (30)
E = {February, April, June, September, November}

(vi) F = {x : x is consonant before k in alphabet}
Consonants: b, c, d, f, g, h, j, (k), l, m, n, p, ...
Consonants before k: b, c, d, f, g, h, j
F = {b, c, d, f, g, h, j}

6 Match each set on the left with the same set on the right

Match the roster form (left) with set-builder form (right):

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

💡 Click to View Solution

1 Understanding Matching

We convert each set-builder form (right) to roster form (left) and then match them.

2 Convert right side to roster form

(a) {x : x is prime and divisor of 6}
Divisors of 6: 1, 2, 3, 6
Prime divisors: 2, 3
In roster form: {2, 3}

(b) {x : x is odd natural number < 10}
Odd natural numbers less than 10: 1, 3, 5, 7, 9
In roster form: {1, 3, 5, 7, 9}

(c) {x : x is natural number and divisor of 6}
Divisors of 6: 1, 2, 3, 6
In roster form: {1, 2, 3, 6}

(d) {x : x is a letter of the word MATHEMATICS}
Letters in MATHEMATICS: M, A, T, H, E, M, A, T, I, C, S
Drop repetitions: M, A, T, H, E, I, C, S
In roster form: {M, A, T, H, E, I, C, S}

3 Matching

Now compare with left side:

(i) {1, 2, 3, 6} matches with (c)
(ii) {2, 3} matches with (a)
(iii) {M, A, T, H, E, I, C, S} matches with (d)
(iv) {1, 3, 5, 7, 9} matches with (b)

✓ Final Matching:
(i) → (c)
(ii) → (a)
(iii) → (d)
(iv) → (b)

📚 Exercise 1.1 - Sets

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