ML Aggarwal Class 7 Maths Solutions Chapter 05 Sets

Question 1. State which of the following collections are sets:
(i) All states of India
(ii) Four cities of India having more than one lakh population
(iii) All tall students of your school
(iv) Four colours of a rainbow
(v) All beautiful flowers
(vi) All clever people of Lucknow
(vii) Last three days of a week
(viii) All months of a year having at least 30 days
Answer: A set must be a well-defined group of items. This means we can clearly say whether any given item belongs to it or not, without any doubt or disagreement.
(i) All states of India - This is clearly defined. We know exactly which places are states and which are not. It forms a set.
(ii) Four cities of India having more than one lakh population - Many cities in India have over one lakh people, so we cannot say for certain which four cities the question refers to. This is not clearly defined. It is not a set.
(iii) All tall students of your school - The word "tall" means different things to different people. There is no clear measure of what counts as tall. It is not a set.
(iv) Four colours of a rainbow - A rainbow shows seven colours, so saying "four colours" is unclear - we do not know which four the question means. It is not a set.
(v) All beautiful flowers - Beauty is a personal matter; what looks beautiful to one person may not to another. It is not a set.
(vi) All clever people of Lucknow - Cleverness cannot be measured the same way for everyone. It is not a set.
(vii) Last three days of a week - This is clear and definite: Friday, Saturday, and Sunday. It forms a set.
(viii) All months of a year having at least 30 days - We can name these months exactly: January, March, May, July, August, October, and December all have 31 days. It forms a set.
In simple words: A set is a group where we can say for sure if something belongs to it. Groups (i), (vii), and (viii) are sets because they are well-defined. Groups (ii), (iii), (iv), (v), and (vi) are not sets because they are not clearly defined.

Exam Tip: Always check whether a collection is well-defined - can you say with certainty whether each item is in it or not? If there is any doubt or it depends on personal opinion, it is not a set.

Question 2. If A = {vowels of English alphabet}, then which of the following statements are true. In case a statement is incorrect, mention why.
(i) c ∈ A
(ii) {a} ∈ A
(iii) a, i, u ∈ A
(iv) {a, u} ∉ A
(v) {a, i, u} ∈ A
(vi) a, b ∈ A
Answer: First, let us write out the set A. The vowels in the English alphabet are a, e, i, o, and u. So A = {a, e, i, o, u}.
(i) c ∈ A - This is false. The letter c is not a vowel; it is a consonant. Therefore, c does not belong to A, written as c ∉ A.
(ii) {a} ∈ A - This is false. {a} is itself a set (containing one element), not a single element. Elements of A are the letters a, e, i, o, u by themselves, not sets. A set cannot be an element of another set in this context.
(iii) a, i, u ∈ A - This is true. All three letters a, i, and u are vowels, so they are all inside set A.
(iv) {a, u} ∉ A - This is true. {a, u} is a set containing two elements, not an element by itself. It is a subset of A, but not an element of A.
(v) {a, i, u} ∈ A - This is false. Like {a} and {a, u}, this is a set, not an element. Sets are not elements of A.
(vi) a, b ∈ A - This is false. While a is a vowel and does belong to A, the letter b is a consonant and does not belong to A. For the statement to be true, both a and b would need to be in A.
In simple words: We must know the difference between an element (like the letter a) and a set (like {a}). Elements go inside the set; sets themselves cannot be elements here. Statements (iii) and (iv) are true; the others are false.

Exam Tip: Pay close attention to the notation: a ∈ A means the element a is in set A, while {a} ∈ A would mean the set {a} is in set A, which is different. Watch for this distinction in questions.

Question 3. Describe the following sets:
(i) {a, b, c, d, e, f}
(ii) {2, 3, 5, 7, 11, 13, 17, 19}
(iii) {Friday, Saturday, Sunday}
(iv) {April, August, October}
Answer:
(i) {a, b, c, d, e, f} - These are the first six letters of the English alphabet.
(ii) {2, 3, 5, 7, 11, 13, 17, 19} - These are all the prime numbers that are less than 20. (A prime number is a number greater than 1 that has no factors except 1 and itself.)
(iii) {Friday, Saturday, Sunday} - These are the last three days of the week.
(iv) {April, August, October} - All three of these month names start with a vowel. This set contains all months of the year whose names begin with a vowel.
In simple words: For each set, find what all the items have in common. That shared property is the description of the set.

Exam Tip: When describing a set, look for a single clear rule or pattern that all elements follow. Your description should make it possible to identify every element in the set without listing them.

Question 4. Write the following sets in tabular form and also in set builder form:
(i) The set of even whole numbers which lie between 10 and 50
(ii) {months of year having more than 30 days}
(iii) The set of single digit whole numbers which are perfect square
(iv) The set of factors of 36
Answer:
(i) The set of even whole numbers which lie between 10 and 50
The even whole numbers from 10 to 50 are: 12, 14, 16, and so on, up to 48.
Tabular form: {12, 14, 16, ..., 48}
Set builder form: {x : x = 2n, n ∈ N and 5 < n < 25}

(ii) {months of year having more than 30 days}
The months with more than 30 days (that is, 31 days) are: January, March, May, July, August, October, and December.
Tabular form: {January, March, May, July, August, October, December}
Set builder form: {x : x is a month of a year having more than 30 days}

(iii) The set of single digit whole numbers which are perfect square
The single digit whole numbers range from 0 to 9. Among these, the perfect squares are 0, 1, 4, and 9.
Tabular form: {0, 1, 4, 9}
Set builder form: {x : x is a perfect square one digit number}

(iv) The set of factors of 36
The factors (or divisors) of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Tabular form: {1, 2, 3, 4, 6, 9, 12, 18, 36}
Set builder form: {x : x is a factor of 36}
In simple words: Tabular form means listing all elements. Set builder form means writing a rule that describes which elements belong to the set.

Exam Tip: In tabular form, always show at least three elements (or use "...") to make the pattern clear. In set builder form, make your rule simple and exact so anyone can identify the elements without guessing.

Question 5. Write the following sets in roster form and also in description form:
(i) {x : x = 4n, n ∈ W and n < 5}
(ii) {x : x = n², n ∈ N and n < 8}
(iii) {y : y = 2x - 1, x ∈ W and x < 5}
(iv) {x : x is a letter in word ULTIMATUM}
Answer:
(i) {x : x = 4n, n ∈ W and n < 5}
Since n is a whole number and n < 5, the possible values are n = 0, 1, 2, 3, 4.
When n = 0: x = 4 × 0 = 0
When n = 1: x = 4 × 1 = 4
When n = 2: x = 4 × 2 = 8
When n = 3: x = 4 × 3 = 12
When n = 4: x = 4 × 4 = 16
Roster form: {0, 4, 8, 12, 16}
Description form: {whole numbers divisible by 4 and less than 20}

(ii) {x : x = n², n ∈ N and n < 8}
Since n is a natural number and n < 8, the possible values are n = 1, 2, 3, 4, 5, 6, 7.
When n = 1: x = 1² = 1
When n = 2: x = 2² = 4
When n = 3: x = 3² = 9
When n = 4: x = 4² = 16
When n = 5: x = 5² = 25
When n = 6: x = 6² = 36
When n = 7: x = 7² = 49
Roster form: {1, 4, 9, 16, 25, 36, 49}
Description form: {squares of the first seven natural numbers}

(iii) {y : y = 2x - 1, x ∈ W and x < 5}
Since x is a whole number and x < 5, the possible values are x = 0, 1, 2, 3, 4.
When x = 0: y = 2(0) - 1 = -1
When x = 1: y = 2(1) - 1 = 1
When x = 2: y = 2(2) - 1 = 3
When x = 3: y = 2(3) - 1 = 5
When x = 4: y = 2(4) - 1 = 7
Roster form: {-1, 1, 3, 5, 7}
Description form: {odd integers which lie between -2 and 8}

(iv) {x : x is a letter in word ULTIMATUM}
The word ULTIMATUM contains the letters U, L, T, I, M, A, T, U, M. Writing each letter only once, we list: U, L, T, I, M, A.
Roster form: {U, L, T, I, M, A}
Description form: {letters in the word ULTIMATUM}
In simple words: Roster form means listing all elements between curly braces. Description form means writing a sentence that explains what the elements are.

Exam Tip: When converting from set builder form to roster form, substitute each allowed value of the variable carefully and compute the result. Do not skip any value or make arithmetic errors.

Question 6. Write the following sets in roster form:
(i) {x : x ∈ N, 5 ≤ x < 10}
(ii) {x : x = 6p, p ∈ I and -2 ≤ p ≤ 2}
(iii) {x : x = n² - 1, n ∈ N and n < 5}
(iv) {x : x - 1 = 0}
(v) {x : x is a consonant in word NOTATION}
(vi) {x : x is a digit in the numeral 11056771}
Answer:
(i) {x : x ∈ N, 5 ≤ x < 10}
Since x is a natural number with 5 ≤ x < 10, the values are x = 5, 6, 7, 8, 9.
{5, 6, 7, 8, 9}

(ii) {x : x = 6p, p ∈ I and -2 ≤ p ≤ 2}
Since p is an integer with -2 ≤ p ≤ 2, the possible values are p = -2, -1, 0, 1, 2.
When p = -2: x = 6(-2) = -12
When p = -1: x = 6(-1) = -6
When p = 0: x = 6(0) = 0
When p = 1: x = 6(1) = 6
When p = 2: x = 6(2) = 12
{-12, -6, 0, 6, 12}

(iii) {x : x = n² - 1, n ∈ N and n < 5}
Since n is a natural number and n < 5, the possible values are n = 1, 2, 3, 4.
When n = 1: x = 1² - 1 = 0
When n = 2: x = 2² - 1 = 3
When n = 3: x = 3² - 1 = 8
When n = 4: x = 4² - 1 = 15
{0, 3, 8, 15}

(iv) {x : x - 1 = 0}
Solving the equation x - 1 = 0, we get x = 1.
{1}

(v) {x : x is a consonant in word NOTATION}
The word NOTATION has the letters N, O, T, A, T, I, O, N. The consonants among these are N, T, T, N. Writing each consonant only once, we get N and T.
{N, T}

(vi) {x : x is a digit in the numeral 11056771}
The numeral 11056771 has the digits 1, 1, 0, 5, 6, 7, 7, 1. Writing each digit only once, we get 1, 0, 5, 6, and 7.
{1, 0, 5, 6, 7}
In simple words: Work through the rule step by step, find all values that fit the conditions, and then list them in a set.

Exam Tip: Always write each element only once in a set, even if it appears multiple times in the original data. Check that you have covered all cases before finalizing your roster.

Question 7. Write the following sets in set builder form:
(i) {1, 3, 5, 7, ..., 29}
(ii) {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
(iii) {1, 4, 9, 16, 25, ...}
(iv) {1/5, 1/6, 1/7, ..., 1/20}
(v) {-16, -8, 0, 8, 16, 24, 32, 40}
(vi) {January, June, July}
Answer:
(i) {1, 3, 5, 7, ..., 29}
The elements are odd natural numbers less than 30.
{x : x is an odd natural number and x < 30}

(ii) {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
The elements are all prime numbers less than 30.
{x : x is a prime number and x < 30}

(iii) {1, 4, 9, 16, 25, ...}
The elements are squares of natural numbers: 1², 2², 3², 4², 5², ... .
{x : x = n², n ∈ N}

(iv) {1/5, 1/6, 1/7, ..., 1/20}
Each element has the form 1/n where n ranges from 5 to 20.
{x : x = 1/n, n ∈ N and 5 ≤ n ≤ 20}

(v) {-16, -8, 0, 8, 16, 24, 32, 40}
The elements are multiples of 8: 8(-2), 8(-1), 8(0), 8(1), 8(2), 8(3), 8(4), 8(5).
{x : x = 8p, p ∈ I and -2 ≤ p ≤ 5}

(vi) {January, June, July}
The month names January, June, and July all begin with the letter J.
{x : x is a month of the year whose name begins with letter 'J'}
In simple words: Find the pattern or rule that connects all the elements. Write a sentence using the set builder notation {x : ...} that captures this rule.

Exam Tip: Look for common features such as arithmetic patterns (odd, even, multiples), mathematical properties (primes, squares, factors), or descriptive traits (letter names, months) that all elements share.

Question 8. If V is the set of vowels in the word COMPETITION, write the given set in (i) description form, (ii) set builder form, (iii) roster form
Answer: The word COMPETITION contains the letters C, O, M, P, E, T, I, T, I, O, N. The vowels among these are O, E, I, I, O. Listing each vowel only once, we get O, E, and I.

(i) Description form: {vowels in the word COMPETITION}

(ii) Set builder form: {x : x is a vowel in the word COMPETITION}

(iii) Roster form: {O, E, I}
In simple words: Pick out only the vowels from the word. In roster form, list them once. In description form, say what they are. In set builder form, write a condition they all follow.

Exam Tip: When identifying vowels, remember they are a, e, i, o, u. Always list each distinct element just once in the final set, even if some letters repeat in the original word.

Question 1. Classify the following sets into empty set, finite set and infinite set. In case of (non-empty) finite sets, mention the cardinal number.
(i) {all colours of a rainbow}
(ii) {x : x is a prime number between 7 and 11}
(iii) {multiples of 5}
(iv) {all straight lines drawn in a plane}
(v) {x : x is a digit in the numeral 550131527}
(vi) {x : x is a letter in word SUFFICIENT}
(vii) {x : x = 4n, n ∈ I and x < 10}
(viii) {x : x ∈ N, x is a prime factor of 180}
(ix) {x : x is a vowel in the word WHY}
(x) {x : x = 5n, n ∈ W and x < 60}
Answer:
(i) {all colours of a rainbow}
A rainbow displays 7 colours: violet, indigo, blue, green, yellow, orange, and red.
Classification: Finite set; Cardinal number = 7

(ii) {x : x is a prime number between 7 and 11}
Numbers between 7 and 11 are 8, 9, and 10. None of these are prime (8 = 2 × 4, 9 = 3 × 3, 10 = 2 × 5).
Classification: Empty set

(iii) {multiples of 5}
Multiples of 5 include 5, 10, 15, 20, 25, ... and continue without end.
Classification: Infinite set

(iv) {all straight lines drawn in a plane}
You can draw as many straight lines as you wish in a plane with no limit.
Classification: Infinite set

(v) {x : x is a digit in the numeral 550131527}
The digits in 550131527 are 5, 5, 0, 1, 3, 1, 5, 2, 7. Listing each digit once: 5, 0, 1, 3, 2, 7.
Classification: Finite set; Cardinal number = 6

(vi) {x : x is a letter in word SUFFICIENT}
The word SUFFICIENT has the letters S, U, F, F, I, C, I, E, N, T. Listing each letter once: S, U, F, I, C, E, N, T.
Classification: Finite set; Cardinal number = 8

(vii) {x : x = 4n, n ∈ I and x < 10}
Since n is an integer and x = 4n must be less than 10, we have 4n < 10, so n < 2.5. Integer values satisfying this: n = ..., -2, -1, 0, 1, 2.
When n = 2: x = 8
When n = 1: x = 4
When n = 0: x = 0
When n = -1: x = -4
When n = -2: x = -8
And smaller values of n give x more negative. The set is {... -8, -4, 0, 4, 8}.
Classification: Infinite set

(viii) {x : x ∈ N, x is a prime factor of 180}
First, find the prime factorization of 180: 180 = 4 × 45 = 4 × 9 × 5 = 2² × 3² × 5.
The prime factors are 2, 3, and 5.
Classification: Finite set; Cardinal number = 3

(ix) {x : x is a vowel in the word WHY}
The word WHY contains only the letters W, H, Y. None of these are vowels (the vowels are a, e, i, o, u).
Classification: Empty set

(x) {x : x = 5n, n ∈ W and x < 60}
Since n is a whole number and x = 5n < 60, we have 5n < 60, so n < 12.
Possible values: n = 0, 1, 2, 3, ..., 11
The set is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55}.
Classification: Finite set; Cardinal number = 12
In simple words: An empty set has no elements. A finite set has a fixed number of elements you can count. An infinite set goes on forever without ending. The cardinal number tells you how many elements are in a finite set.

Exam Tip: When classifying, first decide if the set is empty (no elements at all). If not empty, check whether it has a definite end - if yes, it is finite; if it can grow without limit, it is infinite. For finite sets, always count and state the cardinal number.

Question 3. A set with a limited number of distinct elements is called
(1) a finite set
(2) an infinite set
(3) both finite as well as infinite set
(4) none of these
Answer: (1) a finite set
In simple words: When a set has only a few elements that you can count, it is called a finite set.

Exam Tip: Memorize the difference between finite and infinite sets - finite means countable and limited, infinite means endless.

Question 4. The symbol ↔ stands for
(1) belongs to
(2) is a subset of
(3) is equivalent to

Question 4. Which of the following is correct?
(a) The biconditional symbol ↔ connects two unequal sets.
(b) The biconditional symbol ↔ connects two sets with the same cardinal number.
(c) The biconditional symbol ↔ connects two sets that are identical in every way.
(d) None of the options
Answer: (b) The biconditional symbol ↔ connects two sets with the same cardinal number.
In simple words: Two sets are connected by ↔ when they have the same number of elements, even if the elements themselves are different.

Exam Tip: Remember that ↔ means "equivalent" (same size), not "equal" (same elements). This is a frequently tested distinction.

Question 5. How is the empty set shown?
(1) {φ}
(2) { }
(3) {0}
(4) 0
Answer: (2) { }
In simple words: An empty set has no items in it and is written as two curly brackets with nothing inside. The symbols {φ} and {0} each have one thing inside, so they are not empty.

Exam Tip: Distinguish between the symbol φ (which represents emptiness) and the set {φ} (which contains one element - the symbol itself). Similarly, {0} contains the number zero, so it is not empty.

Question 6. The cardinal number n(A) for A = {x : x is an odd prime number less than 20} is
(1) 8
(2) 7
(3) 9
(4) 10
Answer: (2) 7
In simple words: The odd prime numbers below 20 are 3, 5, 7, 11, 13, 17, and 19. That gives us 7 numbers total. (Note: 2 is prime but it is even, so we do not count it.)

Exam Tip: When finding cardinal numbers, always list all elements carefully and remember that 2 is the only even prime number, so it should be excluded when the question asks for odd primes only.

Question 7. If A = {x | x is a positive multiple of 3 less than 20} and B = {x | x is a prime number less than 20}, then n(A) + n(B) is
(1) 6
(2) 8
(3) 13
(4) 14
Answer: (4) 14
In simple words: Set A contains the numbers 3, 6, 9, 12, 15, 18, which gives n(A) = 6. Set B contains the numbers 2, 3, 5, 7, 11, 13, 17, 19, which gives n(B) = 8. Adding them together: 6 + 8 = 14.

Exam Tip: Work out each set separately and find its cardinal number first, then add the two cardinal numbers together. Double-check your list of primes to avoid missing any or counting incorrectly.

Statement I-II Type Questions

Question 8. Statement I: If A = {x | x ∈ N, x² > 100}, then n(A) = 9 Statement II: Cardinal number of an infinite set is not defined.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (2) Statement I is false but statement II is true.
In simple words: Set A includes all natural numbers whose square is bigger than 100. This means 11, 12, 13, 14, ... and goes on forever, so it is infinite and we cannot count its elements. Statement II is correct - we cannot define a cardinal number for an infinite set.

Exam Tip: When a set continues without end (indicated by "..."), it is infinite, and its cardinal number cannot be defined. Always check if a set is finite before trying to assign a cardinal number.

Question 9. Statement I: If A = {x | x is a colour in the rainbow} and B = {x | x is a vowel in the English alphabet} then n(B) < n(A) Statement II: The cardinal number of a singleton set is 1
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (3) Both Statement I and statement II are true.
In simple words: The rainbow has 7 colours, so n(A) = 7. The vowels are a, e, i, o, u, so n(B) = 5. Since 5 is less than 7, Statement I is true. A singleton set has just one element, so its cardinal number is always 1, making Statement II true as well.

Exam Tip: A singleton set is simply a set with exactly one element. Always remember the vowels in English: a, e, i, o, u (exactly five), and the colours of the rainbow (seven in the traditional classification).

Question 10. Statement I: If A = {x | x ∈ N, x² ≤ 81} and B = {x | x ∈ N, 3x ≤ 27} then A = B Statement II: If A and B are two sets such that A = B, then A ↔ B
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (3) Both Statement I and statement II are true.
In simple words: For set A, x² must be at most 81, so x can be 1, 2, 3, 4, 5, 6, 7, 8, or 9 (since 9² = 81). For set B, 3x must be at most 27, so x can be 1, 2, 3, 4, 5, 6, 7, 8, or 9 (since 3 × 9 = 27). Both sets have the same elements, so A = B. When two sets are equal, they must have the same number of elements, so they are also equivalent (A ↔ B). Both statements are true.

Exam Tip: If A = B, then automatically A ↔ B (equal sets are always equivalent). However, the reverse is not always true - two sets can be equivalent without being equal.

Check Your Progress

Question 1. Write the following sets in tabular form and also in set builder form:
(i) The set of even integers which lie between -6 and 10
(ii) The set of two digit numbers which are perfect square
(iii) {factors of 42}
Answer:
(i) The set of even integers which lie between -6 and 10

The even integers in this range are -4, -2, 0, 2, 4, 6 and 8.

Tabular form: {-4, -2, 0, 2, 4, 6, 8}

Set builder form: {x : x = 2n, n ∈ I and -3 < n < 5}

(ii) The set of two digit numbers which are perfect square

The two digit perfect squares are 16, 25, 36, 49, 64 and 81 (which are 4², 5², 6², 7², 8², 9² respectively).

Tabular form: {16, 25, 36, 49, 64, 81}

Set builder form: {x : x = n², n ∈ N and 4 ≤ n ≤ 9}

(iii) {factors of 42}

The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.

Tabular form: {1, 2, 3, 6, 7, 14, 21, 42}

Set builder form: {x : x is a factor of 42}
In simple words: Tabular form lists all the elements inside curly brackets separated by commas. Set builder form describes what the elements are using a condition or rule, like "x is a factor of 42".

Exam Tip: When converting to tabular form, always list elements in order from smallest to largest. When writing set builder form, make sure the condition you describe actually generates the exact elements shown in tabular form.

Question 2. Write the following sets in roster form:
(i) {x : x = 5n, n ∈ I and -3 < n ≤ 3}
(ii) {x : x = n², n ∈ W and n < 5}
(iii) {x : x = n² - 2, n ∈ W and n < 4}
Answer:
(i) {x : x = 5n, n ∈ I and -3 < n ≤ 3}

The values of n that satisfy the condition are n = -2, -1, 0, 1, 2, 3.

When n = -2, x = 5 × (-2) = -10
When n = -1, x = 5 × (-1) = -5
When n = 0, x = 5 × 0 = 0
When n = 1, x = 5 × 1 = 5
When n = 2, x = 5 × 2 = 10
When n = 3, x = 5 × 3 = 15

Roster form: {-10, -5, 0, 5, 10, 15}

(ii) {x : x = n², n ∈ W and n < 5}

The values of n are n = 0, 1, 2, 3, 4.

When n = 0, x = 0² = 0
When n = 1, x = 1² = 1
When n = 2, x = 2² = 4
When n = 3, x = 3² = 9
When n = 4, x = 4² = 16

Roster form: {0, 1, 4, 9, 16}

(iii) {x : x = n² - 2, n ∈ W and n < 4}

The values of n are n = 0, 1, 2, 3.

When n = 0, x = 0² - 2 = -2
When n = 1, x = 1² - 2 = -1
When n = 2, x = 2² - 2 = 2
When n = 3, x = 3² - 2 = 7

Roster form: {-2, -1, 2, 7}
In simple words: Roster form means you substitute each allowed value into the formula and list all the results you get inside curly brackets.

Exam Tip: Always identify the range of values for the variable first (e.g. which integers n can be), then substitute each one into the given formula to find x, and finally write all the x values in roster form.

Question 3. Write the following sets in set builder form:
(i) {-14, -7, 0, 7, 14, 21, 28}
(ii) {1, 2, 3, 6, 9, 18}
Answer:
(i) {-14, -7, 0, 7, 14, 21, 28}

Looking at these numbers, they are all multiples of 7: 7 × (-2) = -14, 7 × (-1) = -7, 7 × 0 = 0, 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28.

Set builder form: {x | x = 7n, n ∈ I and -2 ≤ n ≤ 4}

(ii) {1, 2, 3, 6, 9, 18}

These numbers are all divisors of 18, meaning each one divides 18 evenly.

Set builder form: {x | x ∈ N, x is a factor of 18}
In simple words: To write set builder form, first look at the elements and find what they have in common. Do they follow a pattern? Are they all multiples of something? Are they all factors of a number? Then describe that common property.

Exam Tip: Check your set builder form by working backwards - use the rule to generate the elements again and confirm you get the original roster form exactly.

Question 4. Classify the following sets into finite set, infinite set and empty set. In case of (non-empty) finite set, mention the cardinal number.
(i) The set of even prime numbers
(ii) {multiples of 9}
(iii) {x : x is a prime factor of 84}
(iv) {x : 2x + 5 = 1, x ∈ N}
(v) {x : x is a month of a year having less than 30 days}
(vi) {x | x is a month of a leap year having 28 days}
Answer:
(i) The set of even prime numbers

The only even prime number is 2, so the set is {2}.

Finite set; cardinal number = 1

(ii) {multiples of 9}

The multiples of 9 are 9, 18, 27, 36, ... and this list goes on forever without ending.

Infinite set

(iii) {x : x is a prime factor of 84}

First, find the prime factorization: 84 = 2² × 3 × 7. The prime factors are 2, 3, and 7.

Finite set; cardinal number = 3

(iv) {x : 2x + 5 = 1, x ∈ N}

Solve the equation: 2x + 5 = 1 gives 2x = -4, so x = -2. Since -2 is not a natural number, there is no element in this set.

Empty set

(v) {x : x is a month of a year having less than 30 days}

Only February has fewer than 30 days (it has 28 or 29 days depending on whether it is a leap year), so the set is {February}.

Finite set; cardinal number = 1

(vi) {x | x is a month of a leap year having 28 days}

In a leap year, February has 29 days, not 28. No other month has exactly 28 days either.

Empty set
In simple words: A finite set has a countable number of elements. An infinite set has elements that never stop. An empty set has no elements at all.

Exam Tip: Be careful with definitions - check whether a condition can actually be satisfied. In part (vi), remember that leap years have 29 days in February, not 28, so the condition cannot be met.

Question 5. In the following, determine whether A and B are equivalent sets, and if so, whether A = B.
(i) A = {1, 3, 5}, B = {Red, Blue, Green}
(ii) A = {prime factors of 70}, B = {prime factors of 60}
(iii) A = {even natural numbers less than 10}, B = {odd natural numbers less than 10}
Answer:
(i) A = {1, 3, 5}, B = {Red, Blue, Green}

Both sets have 3 elements, so n(A) = 3 and n(B) = 3. The sets have the same cardinal number, making them equivalent. However, their elements are different (numbers versus colours), so the sets are not equal.

A ↔ B; A ≠ B

(ii) A = {prime factors of 70}, B = {prime factors of 60}

For set A: 70 = 2 × 5 × 7, so A = {2, 5, 7} and n(A) = 3.
For set B: 60 = 2² × 3 × 5, so B = {2, 3, 5} and n(B) = 3.

Both sets have 3 elements, so they are equivalent. However, 7 is in A but not in B, so the sets are not equal.

A ↔ B; A ≠ B

(iii) A = {even natural numbers less than 10}, B = {odd natural numbers less than 10}

Set A = {2, 4, 6, 8}, so n(A) = 4.
Set B = {1, 3, 5, 7, 9}, so n(B) = 5.

The cardinal numbers are different (4 ≠ 5), so the sets are not equivalent.

A is not equivalent to B
In simple words: Two sets are equivalent when they have the same number of elements (same cardinal number). Two sets are equal when they have the exact same elements. Equivalent does not mean equal - they are different ideas.

Exam Tip: Always check both conditions: first whether n(A) = n(B) for equivalence, and then whether the actual elements are identical for equality. Many students confuse these two concepts.

Question 6. State whether each of the following statement is true or false for the sets A, B and C where A = {x | x ∈ N, x < 40 and x is a multiple of 6} B = {x | x ∈ W, x ≤ 40 and x is a multiple of 8} C = {x | x is a factor of 28}.
(i) A ↔ B
(ii) B ↔ C
(iii) A ↔ C
Answer:
First, identify each set:

A = {x | x ∈ N, x < 40 and x is a multiple of 6} = {6, 12, 18, 24, 30, 36}, so n(A) = 6

B = {x | x ∈ W, x ≤ 40 and x is a multiple of 8} = {0, 8, 16, 24, 32, 40}, so n(B) = 6

C = {x | x is a factor of 28} = {1, 2, 4, 7, 14, 28}, so n(C) = 6

(i) A ↔ B

Since n(A) = 6 = n(B), sets A and B are equivalent.

True

(ii) B ↔ C

Since n(B) = 6 = n(C), sets B and C are equivalent.

True

(iii) A ↔ C

Since n(A) = 6 = n(C), sets A and C are equivalent.

True
In simple words: Two sets are equivalent if they have the same number of elements. Count the elements in each set carefully, then compare their cardinal numbers.

Exam Tip: Remember that 0 is a whole number (W includes 0) but 0 is not a natural number (N does not include 0). This makes a difference when listing the elements of sets defined on N or W.