OP Malhotra Class 11 Maths Solutions Chapter 1 Sets Exercise 1 (A)

S Chand Class 11 ICSE Maths Solutions Chapter 1 Sets Ex 1(a)

Question 1. Which of the collections are sets?
(i) The collection of all months of a year, beginning with letter J
(ii) The collection of most talented writers of India.
(iii) The collection of all natural numbers less than 100.
(iv) A collection of most dangerous animals of the world.
Answer:
(i) This is a well-defined collection of objects (January, June, July), so it is a set. A collection is a set if its members can be clearly identified without any doubt.
(ii) This is not a well-defined collection because the term "most talented" is subjective and can vary from person to person. Therefore, it does not represent a set.
(iii) This is a well-defined collection of numbers (1, 2, 3, ..., 99), so it is a set. All members are clearly defined and can be listed.
(iv) This is not a well-defined collection because "most dangerous" depends on different opinions and criteria. Thus, it does not represent a set.
In simple words: A set is a collection where you can clearly say what belongs and what doesn't. Things like "most talented" or "most dangerous" are not clear, so they don't form sets. Collections of months or specific numbers are clear, so they are sets.

🎯 Exam Tip: Remember that a collection is a set only if its elements are well-defined and distinct. Avoid subjective terms when identifying sets.

Question 2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol \( \in \) and \( \notin \) in the blank spaces.
(i) 5 .... A
(ii) 8 .... A
(iii) 0 .... A
(iv) 4 .... A
Answer: Given A = {1, 2, 3, 4, 5, 6}. We need to check if each number is an element of set A.
(i) Since 5 is a member of set A, we write 5 \( \in \) A. This means 5 belongs to set A.
(ii) Since 8 is not a member of set A, we write 8 \( \notin \) A. This means 8 does not belong to set A.
(iii) Since 0 is not an element of set A, we write 0 \( \notin \) A. This means 0 is not in set A.
(iv) Since 4 is an element of set A, we write 4 \( \in \) A. This means 4 is part of set A.
In simple words: Use \( \in \) if the item is inside the set. Use \( \notin \) if the item is not inside the set.

🎯 Exam Tip: The symbol \( \in \) means "is an element of" or "belongs to," while \( \notin \) means "is not an element of" or "does not belong to."

Question 3. Write down a description of each of the following sets. (There could be different suitable descriptions.)
(i) {2, 4, 6, 8}
(ii) {7, 14, 21, 28, 35}
(iii) {1, 2, 3, 4, 6, 12}
Answer: We need to describe each set using words.
(i) {2, 4, 6, 8} is the set of all even numbers between 1 and 9. These are all whole numbers that can be divided by 2.
(ii) {7, 14, 21, 28, 35} is the set of all multiples of 7 between 1 and 36. Each number in the set is 7 multiplied by a whole number.
(iii) {1, 2, 3, 4, 6, 12} is the set of all factors of 12. These are all the numbers that can divide 12 evenly.
In simple words: We describe what kind of numbers are in each set. For example, even numbers, multiples of 7, or numbers that divide 12.

🎯 Exam Tip: When describing a set, be precise about the properties of its elements and the range they fall within.

Question 4. List the following sets in roster form.
(i) The set of square numbers less than 40.
(ii) The set of colours of the rainbow.
(iii) (a) The set of factors of 144.
(b) The set of prime factors of 144.
(iv) The set of natural numbers less than 50.
(v) The set of consonants in the English alphabet.
(vi) The set of letters in the word ‘Satellite'.
Answer: We will list the elements for each set.
(i) The square numbers less than 40 are \( 1^2=1 \), \( 2^2=4 \), \( 3^2=9 \), \( 4^2=16 \), \( 5^2=25 \), \( 6^2=36 \). So, the set is {1, 4, 9, 16, 25, 36}.
(ii) The set of colours of the rainbow are: {Indigo, Violet, Blue, Green, Yellow, Orange, Red}. (Note: Blue was missing in the source and added to complete the VIBGYOR sequence.)
(iii) (a) To find the factors of 144, we first find its prime factorization. \( 144 = 2^4 \times 3^2 \). The factors are obtained by combining these prime factors in all possible ways. So, the factors are {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 72, 144}.
(b) The prime factors of 144 are the unique prime numbers that divide it. From the prime factorization \( 144 = 2^4 \times 3^2 \), the prime factors are {2, 3}.
(iv) The natural numbers are positive whole numbers. The set of natural numbers less than 50 is {1, 2, 3, 4, 5, ..., 49}. We can represent this by listing the first few and the last one.
(v) The English alphabet has 26 letters. Consonants are all letters except the vowels (A, E, I, O, U). So, the set is {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}.
(vi) The letters in the word ‘Satellite' are s, a, t, e, l, i, t, e. When listing letters in a set, we only include each unique letter once. So, the set is {s, a, t, e, l, i}.
In simple words: To write a set in roster form, you list all its items inside curly brackets, separated by commas. Make sure each item is listed only once.

🎯 Exam Tip: Remember to list only unique elements in a set and ensure all elements meet the given condition. For factors, systematically find all combinations of prime factors.

Question 5. Rewrite the following sets in the indicated notation.
(i) {– 2, – 4, – 6, – 8} ;
(a) Words
(b) Set-builder notation
(ii) Positive multiple of 11 ; Roster form
(iii) {– 9, – 7, – 5, – 3, – 1} ; Set-builder notation
(iv) Even numbers between 27 and 39
(a) Roster form
(b) Set-builder notation
(v) {x: \( 0 < x < 1 \)} ;
(a) Words
(b) Roster form
(vi) (Number line showing dark dots at -3, 1, 5)
(vii) {x: \( x = -3n \) and \( n \in N \)}
(viii) (Number line showing a dark dot at 2)
(ix) {x: \( x = 8 + 2 \)}
(x) {x: \( x < 5 \) or \( 5 < x \leq 10 \)}
(xi) Given \( y = 5x - 2, x \in N \)
(xii) \( \{x | x = \frac{3p+1}{2p-1}, p \in W \text{ and } p \leq 5\} \)
Answer: We will rewrite each set as requested.
(i) {– 2, – 4, – 6, – 8}
(a) Words: The set of all negative even integers. These are whole numbers less than zero that can be divided by 2.
(b) Set-builder notation: \( \{x : x < 0 \text{ and } x \text{ is even}\} \). This notation clearly defines the properties of the elements.

(ii) Positive multiple of 11: These are numbers you get by multiplying 11 by positive whole numbers. The roster form is {11, 22, 33, 44, ...}. The three dots mean the list continues forever.

(iii) {– 9, – 7, – 5, – 3, – 1} is the set of odd integers between -10 and 0. It includes odd numbers from -9 up to -1. The set-builder notation is \( \{x : -9 \leq x \leq -1, x \text{ is odd}\} \).

(iv) Even numbers between 27 and 39
(a) Roster form: These are the even numbers larger than 27 but smaller than 39. So, {28, 30, 32, 34, 36, 38}.
(b) Set-builder notation: \( \{x : 27 < x < 39, x \text{ is even}\} \). This uses inequality signs to show the range.

(v) {x: \( 0 < x < 1 \)}
(a) Words: The set of numbers greater than 0 but less than 1. This includes all fractions and decimals between 0 and 1.
(b) Roster form: This set cannot be written in roster form because there are infinitely many numbers between 0 and 1. We cannot list them all.

(vi) The number line shows dark dots at x = -3, 1, 5. Dark dots mean these specific numbers are included. So, the roster form solution is {-3, 1, 5}.

(vii) {x: \( x = -3n \) and \( n \in N \)}: Here, N stands for natural numbers (1, 2, 3, ...). We substitute \( n = 1, 2, 3, \dots \).
For \( n=1, x = -3(1) = -3 \)
For \( n=2, x = -3(2) = -6 \)
For \( n=3, x = -3(3) = -9 \)
So, the roster form is {-3, -6, -9, ...}.

(viii) The number line shows only one dark dot at x = 2. A dark dot means the number is included. So, the solution in set-builder form is \( \{x | x = 2\} \).

(ix) {x: \( x = 8 + 2 \)}: This is a very simple set where x is equal to 10. The roster form is {10}.

(x) {x: \( x < 5 \) or \( 5 < x \leq 10 \)}: This means all numbers less than 5, or numbers strictly greater than 5 up to 10 (including 10). The interval notation for this set would be \( (-\infty, 5) \cup (5, 10] \). This uses curved brackets for "not included" and square brackets for "included."

(xi) Given \( y = 5x - 2, x \in N \): Here, N means natural numbers (1, 2, 3, ...). We substitute values of x to find y.
When \( x = 1 \implies y = 5(1) - 2 = 3 \in N \)
When \( x = 2 \implies y = 5(2) - 2 = 8 \in N \)
When \( x = 3 \implies y = 5(3) - 2 = 13 \in N \)
And so on. Thus, in roster form, the given set is {3, 8, 13, ...}. This forms a pattern where each number increases by 5.

(xii) Given \( \{x | x = \frac{3p+1}{2p-1}, p \in W \text{ and } p \leq 5\} \): Here, W stands for whole numbers (0, 1, 2, 3, ...). Since \( p \leq 5 \), then \( p = \{0, 1, 2, 3, 4, 5\} \). We substitute these values of p to find x.
When \( p = 0; x = \frac{3(0)+1}{2(0)-1} = \frac{1}{-1} = -1 \)
When \( p = 1; x = \frac{3(1)+1}{2(1)-1} = \frac{4}{1} = 4 \)
When \( p = 2; x = \frac{3(2)+1}{2(2)-1} = \frac{7}{3} \)
When \( p = 3; x = \frac{3(3)+1}{2(3)-1} = \frac{10}{5} = 2 \)
When \( p = 4; x = \frac{3(4)+1}{2(4)-1} = \frac{13}{7} \)
When \( p = 5; x = \frac{3(5)+1}{2(5)-1} = \frac{16}{9} \)
In roster form, the given set is \( \{-1, 4, \frac{7}{3}, 2, \frac{13}{7}, \frac{16}{9}\} \). Each value of p gives a unique x.
In simple words: Roster form means listing all items with commas in curly brackets. Set-builder form uses rules to describe the items. For number lines, dots show exact numbers, lines show ranges.

🎯 Exam Tip: Pay close attention to the type of numbers (natural, whole, integers) and the inequality symbols (\( <, \leq \)) when writing sets in different notations. For set-builder form, clearly define the variable and its properties.

Question 6. State whether each of the following sets is finite or infinite :
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet.
(iii) The set of number which are multiple of 5.
(iv) The set of animals living on earth.<
(v) The set of circles through the origin
(vi) The set of whole numbers greater than 5.
(vii) The set of natural numbers less than one billion.
(viii) The set of integers between – 4 and 4.
(ix) The set of rational numbers between 0 and 1.
Answer: We will determine if each set is finite (countable) or infinite (uncountable).
(i) The set of lines which are parallel to the x-axis is an infinite set. You can draw infinitely many parallel lines to the x-axis, extending indefinitely.
(ii) The set of letters in the English alphabet is a finite set. There are exactly 26 letters, which can be counted.
(iii) The set of numbers which are multiples of 5 is an infinite set. Multiples of 5 (5, 10, 15, ...) continue endlessly and can never be fully counted.
(iv) The set of animals living on earth is a finite set. Although a very large number, the count of all animals is definite and can theoretically be completed.
(v) The set of circles through the origin is an infinite set. You can draw countless circles that all pass through the point (0, 0), each with a different center and radius.
(vi) The set of whole numbers greater than 5 is an infinite set. Whole numbers like 6, 7, 8, ... continue forever without end.
(vii) The set of natural numbers less than one billion is a finite set. This count starts from 1 and goes up to 999,999,999, which is a very large but specific number.
(viii) The set of integers between – 4 and 4 is a finite set. The integers in this range are {-3, -2, -1, 0, 1, 2, 3}, which are exactly 7 integers.
(ix) The set of rational numbers between 0 and 1 is an infinite set. Between any two rational numbers, there are infinitely many other rational numbers, making this an uncountable set.
In simple words: A finite set has a fixed, countable number of items. An infinite set has an endless number of items.

🎯 Exam Tip: To decide if a set is finite or infinite, ask yourself if you could theoretically count all its elements and reach an end. If not, it's infinite.

Question 7. Which of the following sets are empty sets?
(i) A = {x : x is a human being living on Mars}
(ii) B = {x: x is an odd number divisible by 2}
(iii) C = {x : x is a point common to any two parallel lines}
(iv) D = {0}
(v) E = {x : x is a natural number, x < 5 and simultaneously x > 7}
Answer: We need to identify which sets have no elements.
(i) Set A: As of now, there are no human beings living on Mars. So, set A contains no elements, making it an empty set. An empty set is also called a null set.
(ii) Set B: An odd number cannot be divided evenly by 2. Therefore, there is no odd number that is also divisible by 2. Set B has no elements, making it an empty set.
(iii) Set C: Parallel lines never intersect each other. This means they do not have any point in common. So, set C contains no elements, making it an empty set.
(iv) Set D: The set D = {0} contains one element, which is the number 0. Since it has an element, it is not an empty set; it is a singleton set.
(v) Set E: We are looking for a natural number that is both less than 5 and greater than 7 at the same time. No such natural number exists. Therefore, set E contains no elements, making it an empty set.
In simple words: An empty set is a set with nothing in it. If no item can fit the rule for the set, then it's empty.

🎯 Exam Tip: An empty set is represented by \( \emptyset \) or {}. A set containing zero, like {0}, is not an empty set because it contains one element, which is 0.

Question 8. Are the following sets equal? Give reasons.
(i) A = {2, 3} ; B = {x: x is a solution of \( x^2 + 5x + 6 = 0 \)}
(ii) A = {x : x is a letter in the word FOLLOW} ; B = {y: y is a letter in the word WOLF}
Answer: We need to compare the elements of each pair of sets.
(i) Given A = {2, 3}. For set B, we need to find the solutions to the equation \( x^2 + 5x + 6 = 0 \).
We can factorize the quadratic equation:
\( x^2 + 2x + 3x + 6 = 0 \)
\( x(x + 2) + 3(x + 2) = 0 \)
\( (x + 2)(x + 3) = 0 \)
This gives us \( x = -2 \) or \( x = -3 \).
So, set B = {-2, -3}.
Comparing A = {2, 3} and B = {-2, -3}, we see that the elements are different. Therefore, sets A and B are not equal sets. Two sets are equal only if they contain exactly the same elements.

(ii) For set A, the unique letters in the word FOLLOW are F, O, L, W. So, A = {F, O, L, W}.
For set B, the unique letters in the word WOLF are W, O, L, F. So, B = {W, O, L, F}.
Comparing A = {F, O, L, W} and B = {W, O, L, F}, we see that both sets contain the exact same elements, just in a different order. The order of elements in a set does not matter. Therefore, sets A and B are equal sets.
In simple words: Two sets are equal if they have the exact same items inside them. The order of items does not change if sets are equal.

🎯 Exam Tip: To check if sets are equal, ensure every element in the first set is in the second set, and every element in the second set is in the first set. Remember that quadratic equations can have negative solutions.

Question 9. Which of the following are singleton sets?
(i) A = {x; | x | = 5, x \( \in \) N}
(ii) B = {x: \( x^2 - 11x + 24 = 0 \); x \( \in \) N}
(iii) C = {x : \( x^3 = -125 \), x \( \in \) Z}
Answer: We need to find which sets contain exactly one element.
(i) Given A = {x : | x | = 5, x \( \in \) N}. The absolute value |x| = 5 means \( x = 5 \) or \( x = -5 \). Since \( x \in N \) (natural numbers, which are positive), only \( x = 5 \) is a valid solution. So, set A contains only one element, '5'. Therefore, set A is a singleton set. This demonstrates that conditions can limit potential solutions.

(ii) Given B = {x: \( x^2 - 11x + 24 = 0 \); x \( \in \) N}. We solve the quadratic equation:
\( x^2 - 11x + 24 = 0 \)
\( x^2 - 3x - 8x + 24 = 0 \)
\( x(x - 3) - 8(x - 3) = 0 \)
\( (x - 3)(x - 8) = 0 \)
This gives \( x = 3 \) or \( x = 8 \). Both 3 and 8 are natural numbers, so both are valid elements. In roster form, B = {3, 8}. Since set B contains two elements, it is not a singleton set.

(iii) Given C = {x : \( x^3 = -125 \), x \( \in \) Z}. We need to find x such that its cube is -125. We know that \( (-5)^3 = -125 \). So, \( x = -5 \). Since \( x \in Z \) (integers), -5 is a valid solution. Thus, set C contains only one element, '-5'. Therefore, set C is a singleton set.
In simple words: A singleton set has only one item in it. We check how many items fit the rule for each set.

🎯 Exam Tip: Pay close attention to the domain (e.g., Natural Numbers, Integers) specified for x, as it can restrict the number of solutions and determine if a set is singleton.

Question 10. State the value of n (A) for each of the following sets.
(i) A = {Months of the year}
(ii) A = {Planets of our solar system}
(iii) A = {x: x is an integer and \( -8 \leq x \leq 3 \)}
(iv) A = {x : x is an even number}
Answer: The value of n(A) represents the number of elements in set A, also called the cardinal number.
(i) A = {Months of the year}. There are 12 months in a year (January, February, ..., December). So, n(A) = 12. Each month is a distinct element.
(ii) A = {Planets of our solar system}. There are 8 planets in our solar system (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune). So, n(A) = 8. Each planet is a unique element.
(iii) A = {x: x is an integer and \( -8 \leq x \leq 3 \)}. The integers from -8 to 3 (inclusive) are {-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3}. Counting these elements, there are 12 in total. So, n(A) = 12. A simple way to count is \( 3 - (-8) + 1 = 11 + 1 = 12 \).
(iv) A = {x : x is an even number}. Even numbers (..., -4, -2, 0, 2, 4, ...) go on forever in both positive and negative directions. Since the counting of even numbers never ends, this is an infinite set. Therefore, the cardinal number n(A) cannot be determined (it is infinite).
In simple words: n(A) means how many items are in set A. If the set has too many items to count (like infinite), then n(A) cannot be found.

🎯 Exam Tip: For finite sets, count the distinct elements. For infinite sets, state that n(A) cannot be determined. Remember to include both positive and negative integers if the range permits.

Question 11. Use interval notation to represent each set of numbers.
(i) \( -17 < x < 0 \)
(ii) \( 6 \leq x \leq 12 \)
(iii) \( -1 < x \leq 4 \)
(iv) \( -4 \leq x < 7 \)
(v) \( x \leq 3 \) or \( 5 < x \leq 9 \)
(vi) \( \{x | x \geq 99\} \)
(vii) \( \{x | x \neq 1\} \)
(viii) {1, 3, 5, 7, ...}
(ix) \( x \neq 3 \)
Answer: We will convert each set description into interval notation.
(i) For \( -17 < x < 0 \), both -17 and 0 are not included. We use parentheses for non-inclusive endpoints. So, the interval is \( (-17, 0) \).
(ii) For \( 6 \leq x \leq 12 \), both 6 and 12 are included. We use square brackets for inclusive endpoints. So, the interval is \( [6, 12] \).
(iii) For \( -1 < x \leq 4 \), -1 is not included, but 4 is included. So, the interval is \( (-1, 4] \).
(iv) For \( -4 \leq x < 7 \), -4 is included, but 7 is not included. So, the interval is \( [-4, 7) \).
(v) For \( x \leq 3 \) or \( 5 < x \leq 9 \), this means two separate intervals. For \( x \leq 3 \), it is \( (-\infty, 3] \). For \( 5 < x \leq 9 \), it is \( (5, 9] \). The word "or" means we combine them with a union symbol. So, the interval is \( (-\infty, 3] \cup (5, 9] \).
(vi) For \( \{x | x \geq 99\} \), this means all numbers greater than or equal to 99. The upper bound is infinity. So, the interval is \( [99, \infty) \). Infinity always uses a parenthesis.
(vii) For \( \{x | x \neq 1\} \), this means all numbers except 1. This can be written as all numbers less than 1 or all numbers greater than 1. So, the interval is \( (-\infty, 1) \cup (1, \infty) \).
(viii) For {1, 3, 5, 7, ...}, this is the set of odd natural numbers. This set cannot be expressed in standard interval notation because it represents discrete points, not a continuous range. Interval notation is used for continuous ranges of real numbers.
(ix) For \( x \neq 3 \), this means all numbers except 3. Similar to (vii), this can be written as all numbers less than 3 or all numbers greater than 3. So, the interval is \( (-\infty, 3) \cup (3, \infty) \).
In simple words: Interval notation uses brackets to show a range of numbers. Square brackets means the number is included, and round brackets means it's not. \( \infty \) (infinity) always gets round brackets.

🎯 Exam Tip: Remember that square brackets [ ] mean "inclusive" (the endpoint is part of the set), and parentheses ( ) mean "exclusive" (the endpoint is not part of the set). Always use parentheses for infinity ( \( \infty \) or \( -\infty \) ).

Question 12. (Number line representations)
(i) Number line with dark line going left from -1 (dark dot at -1)
(ii) Number line with hollow dot at 2 and dark line going right from 2
(iii) Number line with hollow dots at 1 and 4, and dark line between them
(iv) Number line with dark dots at -1 and 3, and dark line between them
(v) Number line with dark dots at -4 and -1, and dark line between them
Answer: We will interpret each number line and represent it using interval notation.
(i) The dark line starts at \( x = -1 \) (dark dot, meaning -1 is included) and extends continuously to the left (towards negative infinity). So, the solution set is \( (-\infty, -1] \).
(ii) The dark line starts from 2 and goes continuously to the right (towards positive infinity), but there is a hollow dot at \( x = 2 \), meaning 2 is not included. So, the given representation is \( (2, \infty) \).
(iii) There is a dark line between \( x = 1 \) and \( x = 4 \). There are hollow dots at \( x = 1 \) and \( x = 4 \), which means neither 1 nor 4 are included. So, the given representation is \( (1, 4) \).
(iv) There is a dark line from \( x = -1 \) to \( x = 3 \). There are dark dots at \( x = -1 \) and \( x = 3 \), which means both -1 and 3 are included. So, the given representation is \( [-1, 3] \).
(v) There is a dark line from \( x = -4 \) to \( x = -1 \). There are dark dots at \( x = -4 \) and \( x = -1 \), which means both -4 and -1 are included. So, the given representation is \( [-4, -1] \).
In simple words: A dark line shows all numbers in a range. A dark dot means the number is included, like a square bracket. A hollow dot means the number is not included, like a round bracket.

🎯 Exam Tip: Distinguish carefully between dark dots (inclusive, use [ ]) and hollow dots (exclusive, use ( )). Make sure the direction of the line matches the direction of infinity.