Selina Concise Solutions for ICSE Class 7 Mathematics Chapter 13 Set Concepts Some Simple Divisions by Vedic Method

Points to Remember

1. Definition of a Set: A set is a well-defined collection of objects, symbols, or elements. The phrase 'well-defined' means that we can tell with absolute certainty whether any given object belongs to the collection or not. For example, "the set of boys in Class 10 taller than Peter" is well-defined, whereas "the set of tall boys in Class 10" is not well-defined because the term 'tall' is subjective.

2. Elements of a Set: The individual items, symbols, or members that make up a set are called its elements. Sets are usually denoted using capital English letters, with elements written inside curly brackets and separated by commas. For example, \( A = \{5, 10, 12, 15\} \).

3. Use of Symbols: We use the symbol \( \in \) to mean 'belongs to' or 'is an element of'. Conversely, the symbol \( \notin \) means 'does not belong to'.
(i) The order in which elements are listed inside a set does not matter.
(ii) Elements are not repeated within a set; each distinct element is written only once.

4. Representing a Set: Sets can generally be represented in three ways:
(i) Description Method: Explaining the set in words, e.g., "the set of natural numbers".
(ii) Roster or Tabular Method: Listing all elements inside curly braces, e.g., \( N = \{1, 2, 3, 4, 5, \dots\} \).
(iii) Set-builder or Rule Method: Writing a rule or formula that describes the elements, e.g., \( N = \{x : x \in \mathbb{N}\} \).

5. Cardinal Number: The cardinal number represents the total count of elements inside a set. For set \( A \), it is written as \( n(A) \). For example, if \( B = \{2, 4, 6, 8\} \), then \( n(B) = 4 \).

6. Types of Sets:
(i) Finite Set: Contains a countable number of elements.
(ii) Infinite Set: Contains an endless number of elements.
(iii) Empty or Null Set: Contains no elements at all. It is denoted by \( \{\} \) or \( \Phi \). Note that \( n(\Phi) = 0 \). Also, \( \Phi \ne \{0\} \) and \( \{\Phi\} \ne \{0\} \).
(iv) Disjoint Sets: Two or more sets that share no common elements.
(v) Joint or Overlapping Sets: Sets that share at least one element in common.
(vi) Equal Sets: Sets that contain the exact same elements.
(vii) Equivalent Sets: Sets that have the same cardinal number (same count of elements).

Exercise 13(A)

Question 1. Find, whether or not, each of the following collections represent a set:
(i) The collection of good students in your school.
(ii) The collection of the numbers between 30 and 45.
(iii) The collection of fat-people in your colony.
(iv) The collection of interesting books in your school library.
(v) The collection of books in the library and are of your interest.
Answer:
(i) This collection is not a set because the criteria for a student being "good" is subjective and not well-defined.
(ii) This represents a set as the numbers between 30 and 45 are clearly defined.
(iii) This does not form a set because the term "fat" is subjective and lacks a precise definition.
(iv) This is not a set as what makes a book "interesting" varies from person to person.
(v) This collection is a set since you can clearly identify which books align with your personal interests.
In simple words: A collection is only a set if there is no doubt about what is included. Words like 'good', 'fat', or 'interesting' depend on opinion, so they do not form sets.

Exam Tip: Look out for descriptive adjectives like "good", "fat", or "interesting". Since these are subjective, any collection based on them cannot be a set.

Question 2. State whether true or false :
(i) Set {4, 5, 8} is same as the set {5, 4, 8} and the set {8, 4, 5}
(ii) Sets {a, b, m, n} and {a, a, m, b, n, n} are same.
(iii) Set of letters in the word ‘suchismita’ is {s, u, c, h, i, m, t, a}
(iv) Set of letters in the word ‘MAHMOOD’ is {M, A, H, O, D}.
Answer:
(i) True. Changing the order of elements inside a set does not affect its identity.
(ii) True. Repeating elements inside a set does not change the set.
(iii) True. The unique letters are s, u, c, h, i, m, t, and a, as repeating letters are listed only once.
(iv) True. Only the unique letters M, A, H, O, and D are written, ignoring any duplicates.
In simple words: Order and repetition do not matter in sets. As long as the unique elements are identical, the sets are the same.

Exam Tip: Remember that listing an element multiple times inside curly braces has no effect; only distinct elements are counted when evaluating set equality.

Question 3. Let set A = {6, 8, 10, 12} and set B = {3, 9, 15, 18}. Insert the symbol ‘∈’ or ‘∉’ to make each of the following true :
(i) 6 …. A
(ii) 10 …. B
(iii) 18 …. B
(iv) (6 + 3) …. B
(v) (15 - 9) …. B
(vi) 12 …. A
(vii) (6 + 8) …. A
(viii) 6 and 8 …. A
Answer:
(i) \( 6 \in A \) (since 6 is an element of set A)
(ii) \( 10 \notin B \) (since 10 is not an element of set B)
(iii) \( 18 \in B \) (since 18 is an element of set B)
(iv) \( (6 + 3) \in B \) (since \( 6 + 3 = 9 \), and 9 belongs to set B)
(v) \( (15 - 9) \notin B \) (since \( 15 - 9 = 6 \), and 6 does not belong to set B)
(vi) \( 12 \in A \) (since 12 is an element of set A)
(vii) \( (6 + 8) \notin A \) (since \( 6 + 8 = 14 \), and 14 is not in set A)
(viii) \( 6 \text{ and } 8 \in A \) (since both 6 and 8 are elements of set A)
In simple words: Use the symbol \( \in \) if the number is inside the set, and use \( \notin \) if the number is not inside the set. For expressions, solve them first.

Exam Tip: Simplify operations like \( (6 + 3) \) or \( (15 - 9) \) to a single value before checking if they belong to the specified set.

Question 4. Express each of the following sets in roster form :
(i) Set of odd whole numbers between 15 and 27.
(ii) A = Set of letters in the word “CHITAMBARAM”
(iii) B = {All even numbers from 15 to 26}
(iv) P = {x : x is a vowel used in the word ‘ARITHMETIC’}
(v) S = {Squares of first eight whole numbers}
(vi) Set of all integers between 7 and 94; which are divisible by 6.
(vii) C = {All composite numbers between 2 and 20}
(viii) D = Set of Prime numbers from 2 to 23.
(ix) E = Set of natural numbers below 30 which are divisible by 2 or 5.
(x) F = Set of factors of 24.
(xi) G = Set of names of three closed figures in Geometry.
(xii) H = {x : x ∈ W and x < 10}
(xiii) J = {x : x ∈ N and 2x - 3 ≤ 17}
(xiv) K = {x : x is an integer and -3 < x < 5}
Answer:
(i) \( \{17, 19, 21, 23, 25\} \)
(ii) \( A = \{C, H, I, T, A, M, B, R\} \)
(iii) \( B = \{16, 18, 20, 22, 24, 26\} \)
(iv) \( P = \{a, e, i\} \)
(v) \( S = \{0, 1, 4, 9, 16, 25, 36, 49\} \)
(vi) \( \{12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90\} \)
(vii) \( C = \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18\} \)
(viii) \( D = \{2, 3, 5, 7, 11, 13, 17, 19, 23\} \)
(ix) \( E = \{2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28\} \)
(x) \( F = \{1, 2, 3, 4, 6, 8, 12, 24\} \)
(xi) \( G = \{\text{Triangle, Quadrilateral, Circle}\} \)
(xii) \( H = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
(xiii) Solving the inequality:
\( 2x - 3 \le 17 \)
\( \implies 2x \le 20 \)
\( \implies x \le 10 \)
Since \( x \in \mathbb{N} \), we get \( J = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
(xiv) Since \( x \) is an integer between \( -3 \) and \( 5 \), \( K = \{-2, -1, 0, 1, 2, 3, 4\} \).
In simple words: Roster form means listing out every single element that fits the rule, putting them inside curly brackets, and separating them with commas.

Exam Tip: Be careful with boundaries like "between" (excludes endpoints) versus "from... to..." (includes endpoints). Also, ensure whole numbers include 0, whereas natural numbers start at 1.

Question 5. Express each of the following sets in set-builder notation (form) :
(i) {3, 6, 9, 12, 15}
(ii) {2, 3, 5, 7, 11, 13 …. }
(iii) {1, 4, 9, 16, 25, 36}
(iv) {0, 2, 4, 6, 8, 10, 12, …. }
(v) {Monday, Tuesday, Wednesday}
(vi) {23, 25, 27, 29, … }
(vii) {1/3, 1/4, 1/5, 1/6, 1/7, 1/8}
(viii) {42, 49, 56, 63, 70, 77}
Answer:
(i) \( \{x : x \text{ is a natural number divisible by 3 and } x < 18\} \)
(ii) \( \{x : x \text{ is a prime number}\} \)
(iii) \( \{x : x \text{ is a perfect square and } 1 \le x \le 36\} \)
(iv) \( \{x : x \text{ is an even whole number}\} \)
(v) \( \{x : x \text{ is one of the first three days of a week}\} \)
(vi) \( \{x : x \text{ is an odd natural number and } x \ge 23\} \)
(vii) \( \{x : x = \frac{1}{n} \text{ where } n \text{ is a natural number and } 3 \le n \le 8\} \)
(viii) \( \{x : x \text{ is a natural number divisible by 7 and } 42 \le x \le 77\} \)
In simple words: Set-builder notation uses a general variable \( x \) and writes down a rule that describes the property all elements share.

Exam Tip: When writing in set-builder form, specify both the type of number (like natural, whole, prime) and the upper/lower limits to fully define the set.

Question 6. Given : A = {x : x is a multiple of 2 and is less than 25}
B = {x : x is a square of a natural number and is less than 25}
C = {x : x is a multiple of 3 and is less than 25}
D = {x: x is a prime number less than 25}
Write the sets A, B, C and D in roster form.
Answer:
Set \( A \): Multiples of 2 below 25 are even natural numbers:
\( A = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)
Set \( B \): Squares of natural numbers (\( 1^2, 2^2, 3^2, 4^2 \)) below 25:
\( B = \{1, 4, 9, 16\} \)
Set \( C \): Multiples of 3 below 25:
\( C = \{3, 6, 9, 12, 15, 18, 21, 24\} \)
Set \( D \): Prime numbers less than 25:
\( D = \{2, 3, 5, 7, 11, 13, 17, 19, 23\} \)
In simple words: Write down each set as a list of numbers inside curly brackets, choosing only the values that fit the given conditions.

Exam Tip: Remember that natural numbers start from 1, so when listing squares of natural numbers less than 25, 0 is excluded, and 25 itself is excluded since the rule says "less than".

Exercise 13(B)

Question 1. Write the cardinal number of each of the following sets:
(i) A = Set of days in a leap year.
(ii) B = Set of numbers on a clock-face.
(iii) C = {x : x ∈ N and x ≤ 7}
(iv) D = Set of letters in the word “PANIPAT”.
(v) E = Set of prime numbers between 5 and 15.
(vi) F = {x : x ∈ Z and -2 < x ≤ 5}
(vii) G = {x : x is a perfect square number, x ∈ N and x ≤ 30}.
Answer:
(i) A leap year has 366 days, so \( n(A) = 366 \).
(ii) A clock-face has numbers 1 to 12, so \( n(B) = 12 \).
(iii) The set is \( C = \{1, 2, 3, 4, 5, 6, 7\} \), so \( n(C) = 7 \).
(iv) The distinct letters in "PANIPAT" are P, A, N, I, and T, so \( n(D) = 5 \).
(v) The prime numbers between 5 and 15 are 7, 11, and 13, so \( n(E) = 3 \).
(vi) The integers between -2 and 5 (including 5) are -1, 0, 1, 2, 3, 4, and 5, so \( n(F) = 7 \).
(vii) The perfect squares less than or equal to 30 are 1, 4, 9, 16, and 25, so \( n(G) = 5 \).
In simple words: The cardinal number is simply the count of unique elements in the set. You just count them up.

Exam Tip: For letters of a word, never count repeated letters twice. For inequalities like \( -2 < x \le 5 \), note that \( -2 \) is excluded, but \( 5 \) is included.

Question 2. For each set, given below, state whether it is finite set, infinite set or the null set :
(i) {natural numbers more than 100}
(ii) A = {x : x is an integer between 1 and 2}
(iii) B = {x : x ∈ W ; x is less than 100}.
(iv) Set of mountains in the world.
(v) {multiples of 8}.
(vi) {even numbers not divisible by 2}.
(vii) {squares of natural numbers}.
(viii) {coins used in India}
(ix) C = {x | x is a prime number between 7 and 10}.
(x) Planets of the Solar system.
Answer:
(i) Infinite set (since natural numbers continue endlessly past 100).
(ii) Null set (as there are no integers between 1 and 2).
(iii) Finite set (it contains exactly 100 elements, from 0 to 99).
(iv) Finite set (though extremely large, the total number of mountains on Earth is a limited, countable value).
(v) Infinite set (multiples of 8 continue forever).
(vi) Null set (all even numbers are, by definition, divisible by 2).
(vii) Infinite set (since there are infinitely many natural numbers, their squares are also infinite).
(viii) Finite set (the number of different coins officially in use in India is countable).
(ix) Null set (the only integers between 7 and 10 are 8 and 9, neither of which are prime).
(x) Finite set (there is a specific, countable number of planets in our solar system).
In simple words: A set is finite if you can count all its elements and finish. It is infinite if the counting goes on forever. It is a null set if there is nothing inside it.

Exam Tip: Do not mistake large sets (like the number of mountains or planets) for infinite sets. If a set has a definite, countable limit, it is finite.

Question 3. State, which of the following pairs of sets are disjoint :
(i) {0, 1, 2, 6, 8} and {odd numbers less than 10}.
(ii) {birds} and {tress}
(iii) {x : x is a fan of cricket} and {x : x is a fan of football}.
(iv) A = {natural numbers less than 10} and B = {x : x is a multiple of 5}.
(v) {people living in Calcutta} and {people living in West Bengal}.
Answer:
(i) Not disjoint. The set of odd numbers less than 10 is \( \{1, 3, 5, 7, 9\} \). This shares the element 1 with the first set.
(ii) Disjoint. There is no overlapping member between birds and trees.
(iii) Not disjoint. A person can easily be a fan of both cricket and football.
(iv) Not disjoint. Set \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) and set \( B = \{5, 10, 15, \dots\} \). They share the element 5.
(v) Not disjoint. Calcutta is a city in West Bengal, so anyone living in Calcutta also lives in West Bengal.
In simple words: Disjoint sets have absolutely nothing in common. If they share even a single member, they are not disjoint.

Exam Tip: To show that two sets are not disjoint, it is enough to identify a single element that belongs to both sets.

Question 4. State whether the given pairs of sets are equal or equivalent.
(i) A = {first four natural numbers} and B = {first four whole numbers}.
(ii) A = Set of letters of the word “FOLLOW” and B = Set of letters of the word “WOLF”.
(iii) E = {even natural numbers less than 10} and O = {odd natural numbers less than 9}
(iv) A = {days of the week starting with letter S} and B = {days of the week starting with letter T}.
(v) M = {multiples of 2 and 3 between 10 and 20} and N = {multiples of 2 and 5 between 10 and 20}.
(vi) P = {prime numbers which divide 70 exactly} and Q = {prime numbers which divide 105 exactly}
(vii) A = {0², 1², 2², 3², 4²} and B = {16, 9, 4, 1, 0}.
(viii) E = {8, 10, 12, 14, 16} and F = {even natural numbers between 6 and 18}.
(ix) A = {letters of the word SUPERSTITION} and B = {letters of the word JURISDICTION}.
Answer:
(i) Equivalent. Set \( A = \{1, 2, 3, 4\} \) and \( B = \{0, 1, 2, 3\} \). Both have 4 elements, but they are not identical.
(ii) Equal. Set \( A = \{F, O, L, W\} \) and \( B = \{W, O, L, F\} \). They contain the exact same elements.
(iii) Equivalent. Set \( E = \{2, 4, 6, 8\} \) and \( O = \{1, 3, 5, 7\} \). Both sets have 4 elements, but their members are different.
(iv) Equivalent. Set \( A = \{\text{Sunday, Saturday}\} \) and \( B = \{\text{Tuesday, Thursday}\} \). Both have 2 elements, but the days are different.
(v) Equal. Set \( M = \{12, 14, 15, 16, 18\} \) and Set \( N = \{12, 14, 15, 16, 18\} \). Since both contain identical elements, they are equal.
(vi) Equivalent. Set \( P = \{2, 5, 7\} \) and Set \( Q = \{3, 5, 7\} \). Both have 3 elements, but different members.
(vii) Equal. Set \( A = \{0, 1, 4, 9, 16\} \) and Set \( B = \{16, 9, 4, 1, 0\} \). They contain the same elements, just written in a different order.
(viii) Equal. Set \( E = \{8, 10, 12, 14, 16\} \) and Set \( F = \{8, 10, 12, 14, 16\} \). The roster listings contain the exact same elements.
(ix) Neither equal nor equivalent. Set \( A = \{S, U, P, E, R, T, I, O, N\} \) (9 elements) and Set \( B = \{J, U, R, I, S, D, C, T, O, N\} \) (10 elements) have different elements and different cardinalities.
In simple words: Equal sets have the exact same elements inside them. Equivalent sets have the same count of elements, even if the elements themselves are different.

Exam Tip: Equal sets are always equivalent, but equivalent sets are not necessarily equal. Always count the elements to check for equivalence, and inspect the actual elements to check for equality.

Question 5. Examine which of the following sets are the empty sets :
(i) The set of triangles having three equal sides.
(ii) The set of lions in your class.
(iii) {x : x + 3 = 2 and x ∈ N}
(iv) P = {x : 3x = 0}
Answer:
(i) Not empty. Equilateral triangles have three equal sides, so this set contains elements.
(ii) Empty set. There are no real lions present in a classroom.
(iii) Empty set. Solving \( x + 3 = 2 \) gives \( x = -1 \), which is not a natural number.
(iv) Not empty. Solving \( 3x = 0 \) gives \( x = 0 \). The set is \( P = \{0\} \text{ (which contains one element)} \).
Thus, (ii) and (iii) are empty sets.
In simple words: An empty set is one that contains absolutely nothing. If a set contains even a single element like 0, it is not empty.

Exam Tip: Be careful with \( \{0\} \) and \( \Phi \). The set \( \{0\} \) contains the number zero, so its cardinal number is 1. It is not empty.

Question 6. State true or false :
(i) All examples of the empty set are equal.
(ii) All examples of the empty set are equivalent.
(iii) If two sets have the same cardinal number, they are equal sets.
(iv) If n (A) = n (B) then A and B are equivalent sets.
(v) If B = {x : x + 4 = 4}, then B is the empty set.
(vi) The set of all points in a line is a finite set.
(vii) The set of letters in your Mathematics book is an infinite set.
(viii) If M = {1, 2, 4, 6} and N = {x : x is a factor of 12} ; then M = N.
(ix) The set of whole numbers greater than 50 is an infinite set.
(x) If A and B are two different infinite sets, then n (A) = n (B).
Answer:
(i) True. There is only one unique empty set, so all empty sets are identical.
(ii) True. Every empty set has a cardinal number of 0, making them equivalent.
(iii) False. Equivalent sets have the same cardinal number but can contain completely different elements.
(iv) True. Equal cardinal numbers is the definition of equivalent sets.
(v) False. Solving \( x + 4 = 4 \) gives \( x = 0 \), so the set is \( \{0\} \), which is not empty.
(vi) False. A straight line is made up of an infinite number of points.
(vii) False. The total number of letters printed in any physical book is finite and countable.
(viii) False. Factors of 12 are \( \{1, 2, 3, 4, 6, 12\} \), which is not equal to \( M \).
(ix) True. Whole numbers starting from 51 go on endlessly.
(x) False. Different infinite sets can have different types of transfinite cardinalities.
In simple words: Read each statement carefully. An empty set is always unique and has zero elements. Sets are only equal if their exact members match, not just their counts.

Exam Tip: Be alert to "empty set" statements. A set with 0 as its only element, \( \{0\} \), has a cardinal size of 1 and is never empty.

Question 7. Which of the following represent the null set ?
φ, {0}, 0, { }, {φ}
Answer:
Only \( \Phi \) and \( \{\} \) represent the null set. The other symbols do not: \( \{0\} \) is a set with one element (0), 0 is a number and not a set, and \( \{\Phi\} \) is a set containing the null set as an element.
In simple words: The null set has to be completely empty. Only the bare symbol \( \Phi \) or empty curly braces \( \{\} \) mean a null set.

Exam Tip: Never write \( \{\Phi\} \) to represent an empty set. Placing the empty set symbol inside braces makes it a set containing another set, which means its cardinal size is 1.

Exercise 13(C)

Question 1. Fill in the blanks :
(i) If each element of set P is also an element of set Q, then P is said to be …… of Q and Q is said to be of P.
(ii) Every set is a ….. of itself.
(iii) The empty set is a …… of every set.
(iv) If A is proper subset of B, then n (A) …. n (B).
Answer:
(i) subset, superset
(ii) subset
(iii) subset
(iv) less than (<)
In simple words: A subset is a set whose elements all belong to another set. A proper subset must have fewer elements than the parent set.

Exam Tip: Memorize basic set properties: every set is a subset of itself, and the empty set is a subset of all sets.

Question 2. If A = {5, 7, 8, 9} ; then which of the following are subsets of A ?
(i) B = {5, 8}
(ii) C = {0}
(iii) D = {7, 9, 10}
(iv) E = { }
(v) F = {8, 7, 9, 5}
Answer:
(i) \( B \subset A \) (subset, since all its elements 5 and 8 belong to A).
(ii) \( C \not\subset A \) (not a subset, since 0 is not in A).
(iii) \( D \not\subset A \) (not a subset, since 10 is not in A).
(iv) \( E \subset A \) (subset, as the empty set is a subset of every set).
(v) \( F \subset A \) (subset, since it contains the exact same elements as A).
Therefore, (i), (iv), and (v) are subsets of A.
In simple words: For a set to be a subset of A, every single number inside it must also be found inside A. The empty set is always a subset of any set.

Exam Tip: Remember that a set is always a subset of itself. This means set F, which contains all elements of A, is a subset of A.

Question 3. If P = {2, 3, 4, 5} ; then which of the following are proper subsets of P ?
(i) A = {3, 4}
(ii) B = { }
(iii) C = {23, 45}
(iv) D = {6, 5, 4}
(v) E = {0}
Answer:
(i) \( A = \{3, 4\} \) is a proper subset since all its elements are in \( P \), and it is smaller than \( P \).
(ii) \( B = \{\} \) is a proper subset as the empty set is a proper subset of any non-empty set.
(iii) \( C = \{23, 45\} \) is not a proper subset because its elements do not belong to \( P \).
(iv) \( D = \{6, 5, 4\} \) is not a proper subset since 6 is not in \( P \).
(v) \( E = \{0\} \) is not a proper subset because 0 is not an element of \( P \).
Thus, only (i) and (ii) are proper subsets of P.
In simple words: A proper subset must only contain elements from the main set, and it must be strictly smaller than the main set.

Exam Tip: A proper subset cannot be equal to the original set. While a set is a subset of itself, it is not a *proper* subset of itself.

Question 4. If A = {even numbers less than 12}, B = {2, 4}, C = {1, 2, 3}, D = {2, 6} and E = {4}
State which of the following statements are true :
(i) B⊂A
(ii) C⊆A
(iii) D⊂C
(iv) D ⊄ A
(v) E⊇B
(vi) A⊇B⊇E
Answer:
First, list the sets:
\( A = \{2, 4, 6, 8, 10\} \)
\( B = \{2, 4\} \)
\( C = \{1, 2, 3\} \)
\( D = \{2, 6\} \)
\( E = \{4\} \)
Evaluating the statements:
(i) \( B \subset A \): True, all elements of B are in A.
(ii) \( C \subseteq A \): False, elements 1 and 3 are not in A.
(iii) \( D \subset C \): False, 6 is not in C.
(iv) \( D \not\subset A \): False, because both 2 and 6 are in A, meaning \( D \) is a subset of A.
(v) \( E \supseteq B \): False, as B contains elements (like 2) that are not in E.
(vi) \( A \supseteq B \supseteq E \): True, since B is a subset of A, and E is a subset of B.
In simple words: Check if one set is completely contained inside another. The symbol \( \subset \) means "is a subset of", and \( \supseteq \) means "contains as a subset".

Exam Tip: Pay close attention to nested subset relations like \( A \supseteq B \supseteq E \). Break them down into separate parts (\( A \supseteq B \) and \( B \supseteq E \)) to make verification easier.

Question 5. Given A = {a, c}, B = {p, q, r} and C = Set of digits used to form number 1351.
Write all the subsets of sets A, B and C.
Answer:
(i) Set \( A = \{a, c\} \). The subsets are:
\( \Phi, \{a\}, \{c\}, \{a, c\} \)
(ii) Set \( B = \{p, q, r\} \). The subsets are:
\( \Phi, \{p\}, \{q\}, \{r\}, \{p, q\}, \{p, r\}, \{q, r\}, \{p, q, r\} \)
(iii) Set \( C = \{1, 3, 5\} \) (since the digit 1 in 1351 is repeated and only listed once). The subsets are:
\( \Phi, \{1\}, \{3\}, \{5\}, \{1, 3\}, \{1, 5\}, \{3, 5\}, \{1, 3, 5\} \)
In simple words: To find all subsets, write down the empty set first. Then write each element by itself, then pairs of elements, and finally the entire set itself.

Exam Tip: A set with \( n \) elements has \( 2^n \) subsets. Use this formula to check if you have missed any subsets (e.g., \( 2^2 = 4 \) subsets for A, and \( 2^3 = 8 \) subsets for B and C).

Question 6. (i) If A = {p, q, r}, then number of subsets of A = ……
(ii) If B = {5, 4, 6, 8}, then number of proper subsets of B = ……
(iii) If C = {0}, then number of subsets of C = ……
(iv) If M = {x : x ∈ N and x < 3}, then M has …… proper subsets.
Answer:
(i) Since set A has 3 elements, the number of subsets is \( 2^3 = 8 \).
(ii) Set B has 4 elements. The number of proper subsets is \( 2^4 - 1 = 15 \).
(iii) Set C has 1 element, so the number of subsets is \( 2^1 = 2 \).
(iv) Set \( M = \{1, 2\} \), which has 2 elements. The number of proper subsets is \( 2^2 - 1 = 3 \).
In simple words: The total number of subsets is always 2 raised to the power of the number of elements. For proper subsets, subtract 1 from this total because the set itself does not count.

Exam Tip: Remember the formulas: Number of subsets = \( 2^n \), and Number of proper subsets = \( 2^n - 1 \). Always verify the number of elements (\( n \)) first.

Question 7. For the universal set {4, 5, 6, 7, 8, 9, 10, 11, 12, 13} ; find its subsets A, B, C and D such that
(i) A = {even numbers}
(ii) B = {odd numbers greater than 8}
(iii) C = {prime numbers}
(iv) D = {even numbers less than 10}.
Also, find compliments of each set i.e., find A’, B’, C’ and D’.
Answer:
Let the universal set be \( U = \{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\} \).
(i) Subsets:
\( A = \{4, 6, 8, 10, 12\} \)
\( B = \{9, 11, 13\} \)
\( C = \{5, 7, 11, 13\} \)
\( D = \{4, 6, 8\} \)

(ii) Complements (elements in \( U \) that are not in the respective set):
\( A' = \{5, 7, 9, 11, 13\} \)
\( B' = \{4, 5, 6, 7, 8, 10, 12\} \)
\( C' = \{4, 6, 8, 9, 10, 12\} \)
\( D' = \{5, 7, 9, 10, 11, 12, 13\} \)
In simple words: The complement of a set contains all the numbers from the universal set that are missing from that set.

Exam Tip: When finding complements, always double-check against the universal set to make sure you have not omitted any elements or included elements not in \( U \).

Exercise 13(D)

Question 1. If A = {4, 5, 6, 7, 8} and B = {6, 8, 10, 12}, find :
(i) A ∪ B
(ii) A ∩ B
(iii) A - B
(iv) B - A
Answer:
(i) \( A \cup B = \{4, 5, 6, 7, 8, 10, 12\} \) (combine all unique elements from both sets)
(ii) \( A \cap B = \{6, 8\} \) (find only the elements that appear in both sets)
(iii) \( A - B = \{4, 5, 7\} \) (elements of set A with any elements belonging to B removed)
(iv) \( B - A = \{10, 12\} \) (elements of set B with any elements belonging to A removed)
In simple words: Union (\( \cup \)) joins everything together. Intersection (\( \cap \)) keeps only the shared items. Subtraction (\( A - B \)) starts with A and throws away anything that is also in B.

Exam Tip: Be careful with set subtraction: \( A - B \) is not the same as \( B - A \). Always start with the first set and remove any elements that appear in the second set.

Question 2. If A = {3, 5, 7, 9, 11} and B = {4, 7, 10}, find:
(i) n(A)
(ii) n(B)
(iii) A ∪ B and n(A ∪ B)
(iv) A ∩ B and n(A ∩ B)
Answer:
(i) Set A has 5 elements, so \( n(A) = 5 \).
(ii) Set B has 3 elements, so \( n(B) = 3 \).
(iii) \( A \cup B = \{3, 4, 5, 7, 9, 10, 11\} \), which has 7 elements, so \( n(A \cup B) = 7 \).
(iv) \( A \cap B = \{7\} \), which contains 1 element, so \( n(A \cap B) = 1 \).
In simple words: Find the sets first by listing their elements, then simply count how many elements are in each to get the cardinal number.

Exam Tip: Always verify your count using the cardinal identity formula: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \). In this case: \( 5 + 3 - 1 = 7 \), which matches.

Question 3. If A = {2, 4, 6, 8} and B = {3, 6, 9, 12}, find:
(i) (A ∩ B) and n(A ∩ B)
(ii) (A - B) and n(A - B)
(iii) n(B)
Answer:
(i) \( A \cap B = \{6\} \), so \( n(A \cap B) = 1 \).
(ii) \( A - B = \{2, 4, 8\} \), so \( n(A - B) = 3 \).
(iii) Since B has 4 elements, \( n(B) = 4 \).
In simple words: Identify the elements of the required operations, list them inside curly braces, and count them to find the cardinal size.

Exam Tip: Always make sure to write down the set itself first, then write its cardinality separately, as they are two different things requested by the question.

Question 4. If P = {x : x is a factor of 12} and Q = {x: x is a factor of 16}, find :
(i) n(P)
(ii) n(Q)
(iii) Q - P and n(Q - P)
Answer:
First, find the sets in roster form:
\( P = \{1, 2, 3, 4, 6, 12\} \)
\( Q = \{1, 2, 4, 8, 16\} \)

(i) Since set P has 6 elements, \( n(P) = 6 \).
(ii) Since set Q has 5 elements, \( n(Q) = 5 \).
(iii) Removing factors of 12 from factors of 16:
\( Q - P = \{8, 16\} \)
Therefore, \( n(Q - P) = 2 \).
In simple words: Write down all the factors for both numbers. Then count them, and subtract one set from the other by keeping only the factors of 16 that do not divide 12.

Exam Tip: Don't forget that 1 and the number itself are always factors. Omitting them will lead to incorrect cardinal counts.

Question 5. M = {x : x is a natural number between 0 and 8} and N = {x : x is a natural number from 5 to 10}. Find :
(i) M - N and n(M - N)
(ii) N - M and n(N - M)
Answer:
First, find the sets in roster form:
\( M = \{1, 2, 3, 4, 5, 6, 7\} \) (natural numbers between 0 and 8)
\( N = \{5, 6, 7, 8, 9, 10\} \) (natural numbers from 5 to 10)

(i) Removing elements of N from M:
\( M - N = \{1, 2, 3, 4\} \)
Therefore, \( n(M - N) = 4 \).

(ii) Removing elements of M from N:
\( N - M = \{8, 9, 10\} \)
Therefore, \( n(N - M) = 3 \).
In simple words: List out both sets carefully first. For \( M - N \), keep only elements in \( M \) that are not in \( N \). For \( N - M \), keep only elements in \( N \) that are not in \( M \).

Exam Tip: Pay attention to wording: "between 0 and 8" excludes both 0 and 8, whereas "from 5 to 10" includes both 5 and 10.

Question 6. If A = {x: x is natural number divisible by 2 and x < 16} and B = {x: x is a whole number divisible by 3 and x < 18}, find :
(i) n(A)
(ii) n(B)
(iii) A ∩ B and n(A ∩ B)
(iv) n(A - B)
Answer:
First, express the sets in roster form:
\( A = \{2, 4, 6, 8, 10, 12, 14\} \)
\( B = \{0, 3, 6, 9, 12, 15\} \) (note that whole numbers start at 0, and 0 is divisible by 3)

(i) Set A has 7 elements, so \( n(A) = 7 \).
(ii) Set B has 6 elements, so \( n(B) = 6 \).
(iii) The common elements are 6 and 12, so:
\( A \cap B = \{6, 12\} \)
Therefore, \( n(A \cap B) = 2 \).
(iv) Subtracting B from A:
\( A - B = \{2, 4, 8, 10, 14\} \)
Therefore, \( n(A - B) = 5 \).
In simple words: List the numbers that fit each description, count them, find what they share, and subtract B's elements from A.

Exam Tip: Always remember that 0 is a whole number and is divisible by any non-zero number, so it must be included in set B.

Question 7. Let A and B be two sets such that n(A) = 75, n(B) = 65 and n(A ∩ B) = 45, find :
(i) n(A ∪ B)
(ii) n(A - B)
(iii) n(B - A)
Answer:
Using the cardinal formulas:
(i) \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
\( \implies n(A \cup B) = 75 + 65 - 45 = 95 \)

(ii) \( n(A - B) = n(A) - n(A \cap B) \)
\( \implies n(A - B) = 75 - 45 = 30 \)

(iii) \( n(B - A) = n(B) - n(A \cap B) \)
\( \implies n(B - A) = 65 - 45 = 20 \)
In simple words: Use standard addition and subtraction with the counts of elements to find the union and set differences.

Exam Tip: Learn these set cardinal equations by heart as they are extremely useful and let you solve union and intersection problems very quickly.

Question 8. Let A and B be two sets such that n(A) = 45, n(B) = 38 and n(A ∪ B) = 70, find :
(i) n(A ∩ B)
(ii) n(A - B)
(iii) n(B - A)
Answer:
Using the cardinal formulas:
(i) \( n(A \cap B) = n(A) + n(B) - n(A \cup B) \)
\( \implies n(A \cap B) = 45 + 38 - 70 = 13 \)

(ii) \( n(A - B) = n(A) - n(A \cap B) \)
\( \implies n(A - B) = 45 - 13 = 32 \)

(iii) \( n(B - A) = n(B) - n(A \cap B) \)
\( \implies n(B - A) = 38 - 13 = 25 \)
In simple words: Find the intersection size first, then subtract it from the individual set sizes to get the set differences.

Exam Tip: If you are unsure, draw a quick Venn diagram with the intersection value in the center, and write the remaining counts on either side.

Question 9. Let n(A) = 30, n(B) = 27 and n(A ∪ B) = 45, find :
(i) n(A ∩ B)
(ii) n(A - B)
Answer:
(i) Using the union formula:
\( n(A \cap B) = n(A) + n(B) - n(A \cup B) \)
\( \implies n(A \cap B) = 30 + 27 - 45 = 12 \)

(ii) Using the subtraction formula:
\( n(A - B) = n(A) - n(A \cap B) \)
\( \implies n(A - B) = 30 - 12 = 18 \)
In simple words: Find the overlapping elements first, then subtract that number from the total size of set A.

Exam Tip: Remember that \( n(A - B) \) represents elements that are strictly in A only, which is why we subtract the overlap \( n(A \cap B) \) from \( n(A) \).

Question 10. Let n(A) = 31, n(B) = 20 and n(A ∩ B) = 6, find:
(i) n(A - B)
(ii) n(B - A)
(iii) n(A ∪ B)
Answer:
(i) \( n(A - B) = n(A) - n(A \cap B) \)
\( \implies n(A - B) = 31 - 6 = 25 \)

(ii) \( n(B - A) = n(B) - n(A \cap B) \)
\( \implies n(B - A) = 20 - 6 = 14 \)

(iii) \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
\( \implies n(A \cup B) = 31 + 20 - 6 = 45 \)
In simple words: Find the size of the differences by subtracting the overlap from each set's size, and find the union by adding the sizes and subtracting the overlap once.

Exam Tip: A useful alternative formula for union is \( n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B) \). Substituting the values gives: \( 25 + 14 + 6 = 45 \), which verifies the result.