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11M02_Sets, Relations and Functions_Avanti Module

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M2 – Sets, Relations and Functions 39 Sixth Edition M2. Sets, Relations and Functions TABLE OF CONTENTS M2. Sets, Relations and Functions 39 M2.1 Introduction to Sets ...................................................................................................................................................................41 M2.2 Set Operations and Venn Diagrams....................................................................................................................................49 M2.3 Relations and Functions ..........................................................................................................................................................55 M2.4 Domain and Range .....................................................................................................................................................................60 Test Practice Problems .........................................................................................................................................................................64 Answer Key ................................................................................................................................................................................................71

M2 – Sets, Relations and Functions 40 PRE-TEST Look at the table below. Place a tick mark against each number type that matches the number in the first column. Natural Whole Integer Rational Irrational Real None 4 3.89 3. 3̅ −3 √−1 √2 2 3 0 ������ Score 1 point per correct answer. If you score less than 7 then take some time to revise the following topics from NCERT IX Chapter 1.  Real Numbers  Rational and Irrational Numbers  Integers  Natural Numbers and Whole Numbers M2.1

M2 – Sets, Relations and Functions 41 M2.1 Introduction to Sets CONCEPTS 1. Elements of a set. 2. Representation of a set in roster and set-builder form and interconversion from one form to another. 3. Classification of sets into null/empty/void set, singleton set, finite set and infinite set. 4. Equality of two sets. 5. Definition and representation of subsets, proper subsets, supersets, universal sets and power set. 6. Representation of open and closed intervals. PRE-READING Category Book Name Chapter Name Chapter Section Compulsory NCERT Maths Class XI Sets 1 1.2 to 1.8 SYNOPSIS 1. DEFINITION A set is a collection of well-defined objects which are distinct from each other. A set is denoted by { }, with the objects separated by commas. Sets are usually defined by capital letters ������, ������, ������, . .. and elements are usually denoted by ������, ������, ������, … Some examples of sets are: I. Odd numbers less than 10, i.e., {1, 3, 5, 7, 9} II. Various kinds of triangles. III. The vowels in the English alphabet, {������, ������, ������, ������, ������ } IV. The solution of the equation : ������2 − 5������ + 6 = 0, i.e., {2, 3} Some examples of not well defined sets are: I. Set of young people. II. Set of good students in a class. 2. STANDARD NOTATIONS Symbol The set of all ������ → Natural numbers ������ → Whole numbers ������ → Integers ������+/������− → Positive/Negative integers ������ → Real numbers ������+/������− → Positive/Negative real numbers ������ → Rational numbers ������+/������− → Positive/Negative Rational numbers ������ → Irrational numbers

M2 – Sets, Relations and Functions 42 3. REPRESENTATION OF SETS I. Roster or tabular forms : In this method, a set is described by listing all the elements, separated by commas, within braces. For example : The set of all vowels in English alphabet, ������ = {������, ������, ������, ������, ������} ������ ∈ ������ denotes that ������ belongs to ������. ������ ∉ ������ denotes that ������ does not belong to ������. II. Set builder form : In this method, we write down a property or rule which gives us all the elements of the set by that rule. For example : a) ������ = {1, 2, 3, 6, 7, 14, 21, 42} can be represented as ������ = {������ ∶ ������ is a natural number which divides 42} b) ������ = {4, 5, 6, 7, 8, 9} can be represented as ������ = {������ ∶ ������ is a natural number and 3 < ������ < 10} 4. INTERVAL Let ������ and ������ be two real numbers. The set of all real numbers between ������ and ������ form an interval. I. OPEN INTERVAL The set {������ ∈ ������: ������ < ������ < ������} is called an open interval and is denoted by (������, ������). (Here, ������ does not take the end values ������ and ������.) II. CLOSED INTERVAL The set {������ ∈ ������: ������ ≤ ������ ≤ ������} is called an open interval and is denoted by [������, ������]. (Here, ������ takes the end values ������ and ������ in addition to taking the values between ������ and ������.) 5. TYPES OF SETS I. FINITE SET : A set containing finite number of elements or no element. E.g : ������ = {������|������ is an integer, 1 ≤ ������ ≤ 4}, i.e., {1,2,3,4} II. INFINITE SET : A set containing infinite number of elements. E.g : ������ = {������ ∶ ������ ∈ ������, ������ > 4}, i.e., {5, 6, 7, 8, … } III. EMPTY SET (NULL/VOID SET) : A set containing no element, it is denoted by ������ or { }. E.g : ������ = {������ ∶ ������ is a student presently studying in both classes 9������ℎ and 10������ℎ} IV. SINGLETON SET : A set containing a single element. E.g. ������ = {1} V. EQUAL SETS : Two sets ������ and ������ are said to be equal, if every element of ������ is a member of ������ and every element of ������ is a member of ������ and we write ������ = ������. E.g. ������ = {������: ������ is a letter in the word ������������������������������������������������}, ������ = {������, ������, ������, ������, ������, ������, ������} VI. EQUIVALENT SETS : Two sets are said to be equivalent, if they have same number of elements. Number of elements in a set ������ is denoted by ������(������). If ������(������) = ������(������), then ������ and ������ are equivalent sets. But converse is not true. VII. SUBSETS : A set ������ is said to be a subset of ������ if every element of ������ is also an element of ������. We represent it as ������ ⊂ ������. E.g. ������ = {1}, ������ = {1,2,3} here, ������ ⊆ ������. M2.1

M2 – Sets, Relations and Functions 43 Subsets are further divided into two types: a) PROPER SUBSETS : If ������ is a subset of ������, but ������ has atleast one element that is not in ������, then ������ is a strict subset or proper subset of ������. In this case, ������ not equal to ������. Denoted by ������ ⊂ ������. b) IMPROPER SUBSETS : If ������ is a subset of ������ and ������ = ������, then A is called improper subset of ������ and we write ������ ⊆ ������. E.g : ������ = {������, ������, ������}, ������ = {������, ������, ������}. Here ������ ⊆ ������ and ������ ⊆ ������. VIII. POWER SET : The set formed by all the subsets of a given set ������, is called power set of ������, denoted by ������(������). If the given set has ������ elements, then the power set contains 2������ elements. The null set and the set itself are always present in the power set. E.g. If ������ = {1,2}, then power set of ������, i.e. ������(������) = {������, {1}, {2}, {1,2}}. IX. UNIVERSAL SET : A set consisting of all possible elements which occurs under consideration is called a universal set. It is denoted by ������.  POINTS TO REMEMBER : I. Every set ������ is a subset of itself i.e., ������ ⊆ ������. II. The set ������ of rational numbers is a subset of the set ������ of real numbers. i.e., ������ ⊂ ������. III. The null set ������ is a subset of every set. IV. If ������ ⊂ ������ and ������ ⊂ ������, then ������ ⊂ ������. V. The set {������} is not a null set. It is a set containing one element ������. VI. Equal sets are always equivalent but equivalent sets may not be equal. PRE-READING EXERCISE Q1. Let ������ = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: Example: 5 ___ ������ Solution: The symbol ∈ (belongs to) is used when a given element is an element of a set while the symbol ∉ (does not belong to) is used when a given element is not an element of a set. 5 is an element of the set A. So 5 ∈ ������. I. 8 ___������ II. 0 ___ ������ III. 4 ___ ������ Q2. ������ = {natural numbers}; ������ = {2, 4, 6, 8, 10}; ������ = {1, 3, 6, 7, 8}. State whether each of the following are true or false and give reasons for your answer: I. 2 ∈ ������ II. 11 ∈ ������ III. 4 ∉ ������ Q3. Consider the sets ������ (null set), ������ = {1, 3}, ������ = {1, 5, 9}, ������ = {1, 3, 5, 7, 9}. Insert the symbol ⊂ or ⊄ between each of the following pair of sets: Example: ������ ⊂ ������. Solution: The null set ������ is a subset of every set. A set is a subset of another set if every element is present in the other set. Since ������ has no element, it is a subset of every set. I. ������ ___ ������ II. ������ ___ ������ III. ������ ___ ������

M2 – Sets, Relations and Functions 44 Q4. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces for the given pairs of sets: I. {������: ������ is a student of Class XI of your school} ___ {������: ������ is a student of your school} II. {������: ������ is a circle } ___ {������: ������ is a circle of radius 1 cm} III. {������: ������ is a triangle} ____ {������: ������ is a rectangle} IV. {������: ������ is an equilateral triangle} ___ {������: ������ is a triangle} V. {������: ������ is an even natural number} ____ {������: ������ is an integer} IN CLASS EXERCISE IN CLASS EXERCISE 1 Use the following paragraph to answer questions 1 to 3. At a recent wedding, the information we gathered about each wedding guest is presented in the table below. The table gives information regarding whether a person was a guest of the bride or the groom and whether a person brought a gift. Based on this information, please answer the following questions. Name Groom’s Guest Bride’s Guest Brought Gift Ram Yes No No Shyam Yes Yes Yes Mohan No Yes Yes Sohan Yes No Yes Sita Yes Yes Yes Gita No No No Q1. Given that the only information you have about the wedding is in the table above, which of the following collections of people is a well-defined set? Example: All the guests of the bride. Solution: A collection is said to be well-defined if we can definitely decide whether a given particular object belongs to this given collection or not. This collection is well-defined as we can definitely decide whether a person is a guest of the bride or not. I. All the guests of both the bride and groom. II. People who brought a gift. III. People who brought an expensive gift. Q2. Let ������ be the set of all the guests of the groom and ������ be the set of all the guests of the bride. Mark the following as true or false. Example: Ram ∈ ������ Solution: False. Set ������ has Shyam, Mohan and Sita as its elements, as they are guests of the bride. So Ram will not be an element of ������. I. Gita ∉ ������ II. Sita ∈ ������ III. Mohan ∉ ������ IV. ������ ⊂ ������ M2.1

M2 – Sets, Relations and Functions 45 Q3. Answer the following questions on the basis of the information given in the table. Also mark whether the given set is singleton or non-singleton. (A set is said to be singleton if it has only one element, otherwise it is a non- singleton set) Example: How many people are guests of the Bride? Solution: Only Shyam, Mohan and Sita are the guests of the Bride. So there are 3 people who are guests of the Bride. Since, there are more than one person in the set, it is a non-singleton set. I. How many people are guests of the Groom? II. How many people are guests of both? III. How many people are guests of the Bride only? IV. How many people are guests of the Groom only? V. How many people are guests of neither the Bride nor the Groom? VI. How many people are guests of the Groom and have brought a gift? VII. How many people are guests of the Bride and did not bring a gift? VIII. How many people are guests of both the bride and the groom and have brought gifts? Q4. Write the following as intervals: Example: {������: ������ ∈ ������, −4 < ������ ≤ 6} Solution: The end of the interval which is included in the set using ≤ symbol is shown by using square brackets [ and ] and the end which is not included in the set by using < symbol is shown by round brackets ( and ). Here −4 is not included in the set while 6 is included in the set. Thus, the given set is represented by (−4, 6]. I. {������: ������ ∈ ������, −4 ≤ ������ < 6} II. {������: ������ ∈ ������, −4 ≤ ������ ≤ 6} III. {������: ������ ∈ ������, −4 < ������ < 6} IV. {������: ������ ∈ ������, −4 < ������} V. {������: ������ ∈ ������, ������ ≤ 6} IN CLASS EXERCISE 2 Q5. Represent the following sets in both Tabular and Set Builder form: Example: The set of all natural numbers which divide 42. Solution: In tabular form, we have to enumerate all the elements of the set within curly brackets and separated by commas. The natural numbers 1, 2, 3, 6, 7, 14, 21 and 42 divide 42. Thus, the required set is {1, 2, 3, 6, 7, 14, 21, 42} in Tabular form. In Set Builder form, we denote the element by a letter ������ followed by a colon “:” and then specify the characteristic property of the elements of that set. This description is enclosed within curly brackets. Thus, the required set is {������: ������ is a natural number and ������ divides 42} in Set Builder form. I. The set of vowels in the English alphabet. II. All even natural numbers. III. All negative integers. Q6. Mark the following as empty set, singleton set, finite set and/or infinite set. (Each set can be more than one type) I. {������: ������ is a real number and ������2 = 1} II. {������: ������ is a real number and ������2 = −1} III. {������: ������ is a real number and ������ − 9 = 0} IV. {������: ������ is a real number}

M2 – Sets, Relations and Functions 46 Q7. Which of the following is correct for the set ������ = {1, 2, {3, 4}, 5}? Example: {1, 2, 3} ⊂ ������ Solution: False. The set ������ has 4 elements 1, 2, 5 and the set {3, 4}. The set {1, 2, 3} has an element 3 which is not present in the set ������. Hence {1, 2, 3} is not a subset of ������. I. 1 ∈ ������ II. 1 ⊂ ������ III. {1, 2, 5} ⊂ ������ IV. {1, 2, 5} ∈ ������ V. {3, 4} ⊂ ������ VI. {3, 4} ∈ ������ VII. {{3, 4}} ⊂ ������ Q8. Given the sets ������ = {1, 3, 5}, ������ = {2, 4, 6} and ������ = {0, 2, 4, 6, 8}. Can the following be considered a universal set for all the three sets ������, ������ and ������? I. {0, 1, 2, 3, 4, 5, 6} II. ������ (null set) III. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} IV. {1, 2, 3, 4, 5, 6, 7, 8} V. ������ Q9. Given a set ������ = {1, 2, 3}. Using this set, answer the following questions: Example: {1} ∈ ������(������) Solution: True. The power set is the set of all subsets of A, including the empty set and A itself. Each element of ������(������) is therefore actually a set. We know that {1} ⊂ {1, 2, 3}. Thus {1} is an element of ������(������). I. Create the power set of ������. II. Find number of elements in the power set of ������. III. State whether 2 ⊂ ������(������) is true or false, where ������(������) is the power set of ������. IV. State whether {2} ∈ ������(������) is true or false, where ������(������) is the power set of ������. V. State whether {1} ⊂ ������(������) is true or false, where ������(������) is the power set of ������. HOMEWORK LEVEL 1 Q1. Is each of the following a well-defined set? Example: The collection of all letters of the English alphabet. Solution: Yes. A collection is said to be well-defined if we can definitely decide whether a given particular object belongs to this given collection or not. Yes. If we are given a symbol we can determine whether it is one of the 26 letters of English alphabet or not. I. The collection of all tall people. II. The collection of all integers ������ for which 2������ − 9 = 16. III. The collection of all good tennis players. IV. The collection of integers V. The collection of five most renowned mathematicians of the world Q2. Which of the following sets are equal? Example: {0, 1, 2, 3} and {1, 0, 2, 3} Solution: Yes. Changing the order in which the elements are listed does not change the set. Both sets have the same elements 0, 1, 2 and 3, so these two are equal irrespective of the order in which they are listed. I. {0, 1, 2, 3} and {0, 0, 1, 3, 2} II. {������, ������, ������, ������, ������} and { ������, ������, ������, ������, ������, ������, ������, ������} III. {10, 20, 25, 30, 40, … } and {������: ������ is a multiple of 10} IV. {������: ������ is a natural number less than 100 and ������ is a perfect square} and {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} M2.1

M2 – Sets, Relations and Functions 47 Q3. State which of the following statements are true and which are false: I. 37 ∉ {������: ������ has exactly two positive integral factors} II. 28 ∈ {������: the sum of the all positive integral factors of ������ is 2������} III. 7747 ∈ {������: ������ is a multiple of 37} Q4. Convert the following sets from Set builder form to Tabular form: Example: {������ ∶ ������ is a natural number and divisor of 6} Solution: {1, 2, 3, 6}. Only 1, 2, 3, and 6 are divisors of 6. I. {������ ∶ ������ is prime number and divisor of 6} II. {������ ∶ ������ = 2������ − 1, where ������ is a natural number} III. {������ ∶ ������ is a month of a year with less than 30 days} Q5. Convert the following sets from Tabular form to Set Builder form: Example: {2, 4, 6, 8, … } Solution: The given set is the set of even natural numbers. Even natural numbers are products of 2 and a natural number. So {������: ������ = 2������ where ������ ∈ N} is the representation of this set in Set Builder form. I. {5, 10, 15, 20, … } II. {… , −3, −2, −1} III. {… , −3, −2, −1, 0, 1, 2, 3, … } Q6. Find the number of elements in the following sets I. ������ = {������: ������ ∈ ������ and ������2 < 49} II. ������ = {������: ������ ∈ ������ and ������ < 500} III. ������ = set of all factors of 25 IV. ������ = Set of spokes (considered distinct) in the ‘Ashok chakra’ in the Indian National flag V. ������ = Set of all prime numbers less than 25 VI. ������ = Set of all letters in the word ‘calculus’ Q7. Mark these sets as singleton, empty, finite and/or infinite. (A set can be of more than one type) I. Set of neither composite nor prime numbers II. Set of people who appeared for JEE 2014 III. Set of all cars with 100 wheels IV. Set of teeth in your mouth V. Set of hands of a snake VI. Set of multiples of 1 VII. Set of Indian women Prime Ministers till date VIII. Set of points lying on a circle Q8. Write the following sets in Set Builder form: Example: (−2, 10] Solution: Here 10 is included in the set while −2 is not. So {������: ������ ∈ ������, −2 < ������ ≤ 10} is the required representation. I. [−2, ∞) II. (−∞, 10) III. (−∞, ∞) Q9. Find out the intervals represented on the given number lines. I. II. III. IV. V.

M2 – Sets, Relations and Functions 48 LEVEL 2 Q10. Which of the following pair of sets is equal? I. Set of players in the Australian cricket team and set of players in the Indian cricket team II. ������ = {2, 3} and ������ = {������ ∶ ������ is a solution of ������2 + 5������ + 6 = 0} III. ������ = {������ ∶ ������ is a letter in the word FOLLOW} and ������ = {������ ∶ ������ is a letter in the word WOLF} Q11. Which of the following are correct? I. {������, ������} ⊂ { ������: ������ is a vowel in the English alphabet} II. {������} ⊂ {������, ������, ������} III. {������} ∈ {������, ������, ������} IV. {������: ������ is an even natural number less than 6} ⊂ {������ ∶ ������ is a natural number which divides 36} Q12. N is the set of natural numbers, Z is the set of integers, Q is the set of rational numbers, R is the set of real numbers and T is the set of irrational numbers. Which of the following are true? I. ������ ⊂ ������ II. ������ ⊂ ������ III. ������ ∈ ������ IV. ������ ⊄ ������ Q13. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. Example: If ������ ∈ ������ and ������ ∈ ������, then ������ ∈ ������ Solution: False, Let ������ = {1, 2} and ������ = {{1, 2}, 3}. Here 1 ∈ ������ and ������ ∈ ������ but 1 is not an element of ������ I. If ������ ⊂ ������ and ������ ∈ ������, then ������ ∈ ������ II. If ������ ⊂ ������ and ������ ⊂ ������, then ������ ⊂ ������ III. If ������ ⊄ ������ and ������ ⊄ ������, then ������ ⊄ ������ IV. If ������ ∈ ������ and ������ ⊄ ������, then ������ ∉ ������ V. If ������ ⊂ ������ and ������ ∉ ������, then ������ ∉ ������ Q14. Examine whether the following statements are true or false: II. {1, 2, 3} ⊂ {1, 3, 5} IV. {������} ∈ {������, ������, ������} I. {������, ������} ⊄ {������, ������, ������} III. {������} ⊂ {������, ������, ������} Q15. Are the following sets superset of the set {Mumbai, New Delhi, Kolkata, Chennai}? I. Set of all Indian cities II. Set of all Asian cities III. Set of all cities in Maharashtra IV. Set of all Indian states Q16. Given that the universal set, ������ = {5, 6, 7, 8, 9, 10, 11, 12}, for sets ������ and ������. Then, list the elements of the following sets: I. A = {������ ∶ ������ is a factor of 60} II. B = {������ ∶ ������ is a prime number} Q17. How many elements are present in the power set of the following sets? I. {������, ������, ������} II. {������, ������, ������, ������, ������} III. {1, 2, 3, 4, 5, 6, 7, 8} Q18. Make the power set of the set A given by ������ = {car, bike, bicycle} LEVEL 3 Q19. Two finite sets have ������ and ������ elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. Find the values of ������ and ������. M2.1

M2 – Sets, Relations and Functions 49 M2.2 Set Operations and Venn Diagrams CONCEPTS 1. Venn diagrams. 2. Operation on sets (union, intersection, difference and complement of sets) and the properties of operations. 3. De Morgan's Laws. 4. Cardinality (number of elements) of a set. PRE-READING Category Book Name Chapter Name Chapter Section Compulsory NCERT Maths Class XI Sets 1 1.9 to 1.12 SYNOPSIS 1. OPERATIONS OF SETS The following operations on sets are represented by Venn diagrams: I. Union of sets: Given two sets ������ and ������, the union is the set that contains elements or objects that belong to either ������ or to ������ or to both. Symbolically, we write ������ ∪ ������. II. Intersection of sets: Given two sets ������ and ������, the intersection is the set that contains common elements of both the sets. Symbolically, we write ������ ∩ ������. If ������ ∩ ������ = ������, then the sets are called Disjoint Sets III. Complement of a set: The complement of a set is the set of all elements in the universal set that are not in the initial set. Symbolically, we write ������′ = ������ − ������

M2 – Sets, Relations and Functions 50 IV. Difference of sets: The difference of set B from set A is the set of all element in A, but not in B. Symbolically, we write ������ − ������. V. Symmetric difference : (������ − ������) ∪ (������ − ������) Symbolically, we write ������∆������ 2. PROPERTIES OF THE OPERATION OF SETS I. Commutative law → ������ ∪ ������ = ������ ∪ ������ → ������ ∩ ������ = ������ ∩ ������ II. Associative law → (������ ∪ ������) ∪ ������ = ������ ∪ (������ ∪ ������) → (������ ∩ ������) ∩ ������ = ������ ∩ (������ ∩ ������) III. Law of identity element, ������ → ������ ∪ ������ = ������ → ������ ∩ ������ = ������ IV. Idempotent law → ������ ∪ ������ = ������ → ������ ∩ ������ = ������ V. Distributive law → ������ ∩ (������ ∪ ������) = (������ ∩ ������) ∪ (������ ∩ ������) VI. Complement laws → ������ ∪ ������′ = ������ → ������ ∩ ������′ = ������ VII. De Morgan’s law → (������ ∪ ������)′ = ������′ ∩ ������′ → (������ ∩ ������)′ = ������′ ∪ ������′ VIII. Law of double complementation: → (������′)′ = ������ M2.2

M2 – Sets, Relations and Functions 51 IX. Laws of empty set and universal set → ������′ = ������ → ������′ = ������ → ������ ∪ ������ = ������ 3. NUMBER OF ELEMENTS IN SETS If ������ and ������ are finite sets, then I. ������(������ ∪ ������) = ������(������) + ������(������) − ������(������ ∩ ������) If ������, ������ and ������ are finite sets, then II. ������(������ ∪ ������ ∪ ������) = ������(������) + ������(������) + ������(������) − ������(������ ∩ ������) − ������(������ ∩ ������) − ������(������ ∩ ������) + ������(������ ∩ ������ ∩ ������) PRE-READING EXERCISE Q1. Let there be sets ������, ������ and ������ such that ������ = {1, 2, 3}, ������ = {2, 3, 4} and ������ = {3, 4, 5}. Given that the universal set for these sets is given by U = {������: ������ ≤ 10, ������ ∈ ������} . Match the columns. Column 1 Column 2 {4, 5, 6, 7, 8, 9, 10} A) ������ ∪ ������ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B) ������ ∩ ������ {1, 2, 6, 7, 8, 9, 10} C) ������′ {1, 2, 3, 4, 5} D) ������ − ������ {3} E) ������′ {1, 2, 3, 4} F) ������ ∩ ������ {3, 4} G) ������ ∪ ������′ {4} H) ������ − ������ {2, 3} I) ������ ∩ ������ ∩ ������ {1} J) ������ ∪ ������ ∪ ������ IN CLASS EXERCISE IN CLASS EXERCISE 1 Q1. Let ������ = {������: ������ is a vowel of English alphabet} and ������ = {������, ������, ������}. Find the following sets. Example: ������ ∪ ������ Solution: ������ = {������, ������, ������, ������, ������} and ������ = {������, ������, ������}, so ������ ∪ ������ will have elements present in either ������ or ������ or both. The elements present in ������ are ������, ������, ������, ������ and ������ while elements present in ������ are ������, ������ and ������. So ������ ∪ ������ = {������, ������, ������, ������, ������, ������, ������} (The element ������ is present in both ������ and ������, but repetition is not required) I. ������ ∩ ������ II. ������ − ������ III. ������’ (where ������ is the set of all letters of English alphabet)

M2 – Sets, Relations and Functions 52 Q2. Let ������ be any set such that it lies in the universal set ������. Using this information, match the columns: (More than one option in column 1 can match with options in column 2) Column 1 Column 2 A) ������ ∪ ������ 1. ������ B) ������ ∪ ������ 2. ������ C) ������ ∪ ������ 3. ������ D) ������ ∩ ������ 4. ������′ E) ������ ∩ ������ F) ������ ∩ ������ G) ������ ∪ ������′ H) ������ ∩ ������′ I) (������′)′ J) ������′ K) ������′ IN CLASS EXERCISE 2 Q3. Sketch Venn diagrams that show the universal set ������, the sets ������ and ������, and a single element ������ in each of the following cases: I. ������ ∈ ������; ������ ⊂ ������ II. ������ ∈ ������; ������ and ������ are disjoint III. ������ ∈ ������; ������ ∉ ������; ������ ⊂ ������ IV. ������ ∈ ������; ������ ∈ ������; ������ is not a subset of ������; ������ is not a subset of ������ Q4. In a group of people, 40 like cricket, 35 like tennis and 10 like both cricket and tennis. Then, how many like any game (cricket or tennis)? Q5. Use Venn diagrams to prove: II. ������ ∩ (������ ∪ ������) = (������ ∩ ������) ∪ (������ ∩ ������) I. ������ ∪ (������ ∩ ������) = (������ ∪ ������) ∩ (������ ∪ ������) Note: The statements above are known as distributive laws of union and intersection. Q6. Let there be two sets A and B such that the set ������ has 28 elements, set ������ has 32 elements and ������ ∪ ������ has 50 elements. I. Find the number of elements in ������ ∩ ������, represented as ������(������ ∩ ������). Hint: ������(������ ∪ ������) = ������(������) + ������(������) − ������(������ ∩ ������) II. How many elements does ������ − ������ have? III. How many elements does the set ������ − ������ have? IV. ������(������ ∪ ������) = ������(������ − ������) + ������(������ ∩ ������) + ������(������ − ������) A) True B) False Q7. In a class 60% passed their Physics examination and 58% passed in Mathematics. At least what percentage of students passed both their Physics and Mathematics examination? A) 18% B) 17% C) 16% D) 2% M2.2

M2 – Sets, Relations and Functions 53 IN CLASS EXERCISE 3 Q8. In a survey of 60 people, it was found that 25 people read the Hindustan Times (HT), represented by set ������, 26 read the Times of India (ToI), represented by set ������ and 26 read the Indian Express (IE), represented by set ������. 11 read both HT and ToI, 8 read both ToI and InEx, 9 read both HT and InEx, 3 read all three newspapers. Find the number of people who read at least one of the newspapers. HOMEWORK LEVEL 1 Q1. If ������ = {3, 6, 9, 12, 15, 18, 21}, ������ = {4, 8, 12, 16, 20}, ������ = {2, 4, 6, 8, 10, 12, 14, 16}, ������ = {5, 10, 15, 20}; find the following: Example: ������ − ������ Solution: The set ������ − ������ contains the elements which are present in ������ but not in ������. We notice that out of all the elements in A, 12 is present in both ������ and ������, so ������ − ������ cannot contain 12. Thus, ������ – ������ = {3, 6, 9, 15, 18, 21} I. ������ − ������ II. ������ − ������ III. ������ − ������ Q2. Taking the set of natural numbers as the universal set, write down the complements of the following sets: I. {������: ������ is an odd natural number} II. {������: ������ is an even natural number} III. {������: ������ is a positive multiple of 3} IV. {������: ������ is a prime number} V. {������: ������ is a natural number divisible by 3 and 5} VI. {������: ������ is a perfect square} VII. {������: ������ is a perfect cube} VIII. {������: ������ + 5 = 8 } IX. {������: 2������ + 5 = 9} X. {������: ������ ≥ 7} Q3. If ������ = {3, 5, 7, 9, 11}, ������ = {7, 9, 11, 13}, ������ = {11, 13, 15} and ������ = {15, 17} find Example: ������ ∩ ������ ∩ ������ Solution: ������ ∩ ������ ∩ ������ is a set which contains elements which are present in all the three sets ������, ������ and ������ . The element 11 is the only element present in all three sets. So the set ������ ∩ ������ ∩ ������ = {11} I. ������ ∩ (������ ∪ ������) II. (������ ∩ ������) ∩ (������ ∪ ������) III. (������ ∪ ������) ∩ (������ ∪ ������) LEVEL 2 Q4. Consider the Venn diagram below. Match the condition given in column 1 with the result given in column 2: Column 1 Column 2 A) Region iii is empty ������ ⊂ ������ B) Region ii is empty ������ and ������ are disjoint C) Region iv is empty ������ ∪ ������ = ������ D) Region i is empty ������ ⊂ ������

M2 – Sets, Relations and Functions 54 Q5. In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. If A is the set of mathematics teachers and B is the set of Physics teachers, I. How many teachers are in ������ ∪ ������? II. How many teachers are in ������ ∩ ������? III. How many teachers teach physics? Q6. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages? Q7. Given ������ = {1, 2, 3, 4}, ������ = {3, 4, 5, 6} and ������ = {1, 3, 5} Verify that ������ − (������ ∪ ������) = (������ − ������) ∩ (������ − ������) Q8. A survey shows that 63% of the people watch news channel ‘A’ whereas 76% watch news channel ‘B’. If ������% of the people watch both channel, and everyone surveyed watches at least one news channel, then A) ������ = 35 B) ������ = 63 C) 39 ≤ ������ ≤ 63 D) ������ = 39 Q9. There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C1, 50 to chemical C2, and 30 to both the chemicals C1 and C2. (Individuals can also have a skin disorder without being exposed) I. Find the number of individuals in the universal set of individuals for whom we have data. II. Find the number of individuals exposed to chemical C1 but not to chemical C2 III. Find the number of individuals exposed to chemical C2 but not to chemical C1 IV. Find the number of individuals exposed to either chemical Q10. ������, ������, ������ are three sets such that ������(������) = 25, ������(������) = 20 , ������(������) = 27, ������(������ ∩ ������) = 5 , ������(������ ∩ ������) = 7 and ������ ∩ ������ = ������ then ������(������ ∪ ������ ∪ ������) is equal to A) 60 B) 65 C) 67 D) 72 Q11. Give examples of sets ������, ������ and ������ such that ������ ∩ ������, ������ ∩ ������ and ������ ∩ ������ are non-empty sets and ������ ∩ ������ ∩ ������ = ������. LEVEL 3 Q12. Use Venn diagrams to prove that (������ ∩ ������ ∩ ������)’ = ������’ ∪ ������’ ∪ ������’. Q13. Each student in a class of 40 plays at least one indoor game from among chess, carrom and scrabble. 18 play chess, 20 play scrabble and 27 play carrom. 7 play chess and scrabble, 12 play scrabble and carrom and 4 play chess, carrom and scrabble. Find the number of students who play I. chess and carrom II. chess and carrom but not scrabble. Q14. For three sets P, Q and R with a Universal Set U, identify the given expressions as regions 1 to 8 I. ������’ ∩ ������’ ∩ ������’ PR U II. ������ ∩ ������’ ∩ ������’ 678 III. ������ ∩ ������ ∩ ������’ 5 IV. ������ ∩ ������ ∩ ������’ 34 V. ������ ∩ ������ ∩ ������ VI. ������ ∩ ������’ ∩ ������’ 2 VII. ������ ∩ ������ ∩ ������’ 1Q VIII. ������ ∩ ������’ ∩ ������’ M2.2

M2 – Sets, Relations and Functions 55 M2.3 Relations and Functions CONCEPTS 1. Cartesian product of two sets. 2. Definition and representation of a relation between two sets. 3. Image, domain, co-domain and range for a relation. 4. Definition of function as a relation where every element has exactly one image. 5. Image, pre-image, domain, co-domain and range for a function. 6. Identification of a given relation as a function (vertical line test). PRE-READING Category Book Name Chapter Name Chapter Sections Compulsory NCERT Maths Class XI Relations and Functions 2 2.1 to 2.4 (2.4.1 onwards excluded for this topic) SYNOPSIS 1. ORDERED PAIR: An ordered pair consists of two objects or elements in a given fixed order. Example: (������, ������) is not same as (������, ������) Equality of Ordered Pairs : Two ordered pairs (������1, ������1) and (������2, ������2) are equal, iff ������1 = ������2 and ������1 = ������2. 2. CARTESIAN PRODUCT OF SETS: Consider two sets ������ and ������ where ������ = {1, 2} and ������ = {3, 4, 5}. Set of all ordered pairs of elements of ������ and ������ is {(1,3), (1,4), (1,5), (2,3), (2,4), (2,5)}. This set is denoted by ������ × ������ and is called the Cartesian product of sets ������ and ������. i.e. ������ × ������ = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)} Cartesian product of sets ������ and ������ is denoted by ������ × ������. In the set builder form : ������ × ������ = {(������, ������): ������ ∈ ������ and ������ ∈ ������} ������ × ������ = {(������, ������) ∶ ������ ∈ ������ and ������ ∈ ������ } Properties of Cartesian Product of Sets: ������ × ������ = {(������, ������): ������ ∈ ������ and ������ ∈ ������}. If ������, ������ and ������ are three sets then, I. ������ × (������ ∪ ������) = (������ × ������) ∪ (������ × ������) II. ������ × (������ ∩ ������) = (������ × ������) ∩ (������ × ������) III. ������ × (������ − ������) = (������ × ������) − (������ × ������) IV. If ������ ⊆ ������,then (������ × ������) ⊆ (������ × ������) V. If ������ ⊆ ������, then (������ × ������) ∩ (������ × ������) = ������2 VI. If ������ ⊆ ������ and ������ ⊆ ������ then ������ × ������ ⊆ ������ × ������ VII. (������ × ������) ∩ (������ × ������) = (������ ∩ ������) × (������ ∩ ������), where ������ and ������ are two sets. VIII. If ������ × ������ set has ������ elements, then number of possible subsets = 2������.

M2 – Sets, Relations and Functions 56 3. RELATIONS: A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as If ������ is a relation from ������ to ������ then ������ ⊆ ������ × ������ i.e. , ������ ⊆ {(������, ������): ������ ∈ ������, ������ ∈ ������} Number of possible relations for, ������ to ������ = 2������������ [if the number of elements in ������ is ������ and the number of elements in ������ is ������] 4. DOMAIN OF A RELATION: The domain of a relation is the set of the first element from the ordered pairs. Domain of ������ = {������: (������, ������) ∈ ������} 5. RANGE OF A RELATION: The range of a relation is the set of the second element from the ordered pairs. Range of ������ = {������: (������, ������) ∈ ������} If ������ is a relation from ������ to ������, Then Dom(������) ⊆ ������ and Range (������) ⊆ ������ PRE-READING EXERCISE Q1. Let ������ = {1, 2}, ������ = {1, 2, 3, 4}, ������ = {5, 6} and ������ = {5, 6, 7, 8}. I. Find the sets ������ × ������, ������ × ������ and ������ × ������. II. Is ������ × ������ a subset of ������ × ������? Q2. Which of the following are true? Example: If ������ = {������, ������} and ������ = {������, ������} then ������ × ������ = {(������, ������), (������, ������)} Solution: False. ������ × ������ has elements (������, ������), (������, ������), (������, ������), (������, ������). Here (������, ������) and (������, ������) are missing. I. If ������ = {������, ������} and ������ = ������ then ������ × ������ has no element. II. If ������ = {������, ������} and ������ has infinite elements then ������ × ������ has infinite elements. III. If ������ has ������ elements and ������ has ������ elements, then ������ × ������ has ������������ elements. IV. If ������ = {������, ������} and ������ = {������, ������} then ������ = {(������, ������), (������, ������)} is a relation on ������ → ������ V. If ������ = {������, ������} and ������ = {������, ������} then ������ = {(������, ������), (������, ������)} is a function on ������ → ������ VI. If ������ = {������, ������} and ������ = {������, ������} then ������ = {(������, ������), (������, ������)} is a function on ������ → ������ Q3. Let there be a function, ������, defined on ������ → ������, given by {(1, 2), (3, 5), (4, 7)}. Find I. image of 3 II. pre-image of 7 III. range of ������ M2.3

M2 – Sets, Relations and Functions 57 IN CLASS EXERCISE IN CLASS EXERCISE 1 Q1. Find the values of ������ and ������ if the following ordered pairs are equal. ������ 2 5 1 ������������������������������������������: (3 + 1, ������ − 3) and (3 , 3) Solution: Ordered pairs are equal if their corresponding elements are equal. ������ 5 Thus, 3 + 1 = 3 so ������ = 2. 21 Similarly, ������ − 3 = 3 so ������ = 1. I. (5������, 3������) and (15, 15) II. (������ + 2 , ������ + 3) and (2, 3) III. (������ + ������, ������ − ������ ) and (3, 1) Q2. Cartesian product ������ × ������ consists of 6 elements. If three of these are (1, 2), (2, 3) and (3, 3), then find Cartesian product set ������ × ������ Q3. If ������ = {2, 4, 6, 9} and ������ = {4, 6, 18, 27, 54}, ������ ∈ ������, ������ ∈ ������ I. Find ������(������ × ������) II. Find all elements of ������ × ������ such that ������ is a factor of ������ IN CLASS EXERCISE 2 Q4. Let ������ = {������, ������, ������} and ������ = {1, 2}. Find the number of relations from ������ to ������. Q5. Let ������ = {1, 2, 3, … , 14}. Let there be a relation ������, from ������ to ������, given by ������ = {(������, ������): 3������ − ������ = 0, ������, ������ ∈ ������}. I. Write ������ in roster form II. Represent ������ by arrow diagram III. Find the domain. IV. Find the codomain of ������ V. Find the range of ������ Q6. Let ������ be a relation such that ������ = {(������, ������2): ������ is a prime number less than 10}. I. Write ������ in roster form II. Find the domain of ������ III. Find the range of ������ IN CLASS EXERCISE 3 *Q7. We know that a relation from set ������ to set ������ is a set of ordered pairs, (������, ������), where ������ ∈ ������ and ������ ∈ ������. Since each element belonging to ������ is a pair of numbers ������ and ������, we can represent all the ordered pairs in ������ on the coordinate plane as points. This gives us a collection of points on the coordinate plane, and this collection represents ������. For some relations, this collection might also be a continuous curve, where each point (������, ������) is in the relation. Since a function is also a relation, it will also be possible for us to represent functions on graphs. Using the conditions that are required for a relation to be a function, we can also determine whether any given graph represents a function. (Each element in the domain should have a unique image) Use this information to answer this question:

M2 – Sets, Relations and Functions 58 Which of these graphs represents ������ as a function of ������? (More than one options might be correct) ������ ������ A) ������ B) ������ ������ ������ C) ������ D) ������ Q8. Which of the given relations are functions? Give reasons for your answer. I. ℎ = {(4, 6), (3, 9), (−11, 6), (3, 11)} II. ������ = {(������, ������) | ������ is a real number} IV. ������ = {(������, ������2)| ������ is a natural number} III. ������ = {(������ , 1) | ������ is a natural number} ������ V. ������ = {(������, 3)| ������ is a real number} Q9. Let there be 2 functions ������ and ������ such that ������(������) = ������������ and ������(������) = ln ������. Then prove that (������(1) − ������(1)) × (������(0) − ������(������)) = 0 Q10. If ������(������) = ������−1, then show that ������+1 II. ������ (− 1) = −1 ������(������) I. ������ (1) = −������(������) ������ ������ LEVEL 1 HOMEWORK Q1. Find ������ and ������ if: II. (������ − ������, ������ + ������) = (6, 10) I. (4������ + 3, ������) = (3������ + 5, −2) M2.3

M2 – Sets, Relations and Functions 59 Q2. We have a set of classes in a primary school ������ = {������, ������������, ������������������, ������������, ������} and a set of students who scored more than 90% marks ������ = {������������������, ������ℎ������������������, ������������ℎ������������, ������������ℎ������������, ������������������������, ������������������������} I. Consider the ordered pairs (������������, ������ℎ������������������) and (������ℎ������������������, ������������). Is (������������, ������ℎ������������������) same as (������ℎ������������������, ������������)? II. Write the Cartesian product ������ × ������. III. Write the Cartesian product ������ × ������. IV. Which Cartesian product do the following ordered pairs belong to? a) (������������, Mohan) b) (Sita, Gita) c) (������������, ������������) V. Which of the following are possible relations from ������ to ������? a) {(������, Ram), (������������, Shyam), (������������, Mohan), (Sita, ������������), (������, Sita), (������������������, Gita)} b) {(������, Ram), (������������, Shyam), (������������, Mohan), (������������, ������������), (������, Sita), (������������������, Gita)} c) {(������, Ram), (������������, Shyam), (������������, Mohan), (������������������, Sohan), (������, Sita), (������������������, Gita)} VI. Find the domain, codomain and range for the relation ������ = {(������, Ram), (������������, Shyam), (������������, Mohan), (������, Sita), (������������������, Gita)} Q3. Find ������ and ������ if (Note: ������ × ������ = {(������, ������), (������, ������), (������, ������), (������, ������)} Ordered pairs in ������ × ������ have first element from ������ and second element from ������ . The set of all first elements is ������ which is {������, ������}. Similarly the set of all second elements is ������ which is {������, ������}) I. ������ × ������ = {(1,3), (2,1), (2,3), (1,1)} II. ������ × ������ = {(good, morning), (bad, evening), (good, evening), (bad, morning)} III. (������, 1), (������, 2), (������, 1) are in ������ × ������ and ������ has 3 element and ������ has 2 elements Q4. The Cartesian product ������ × ������ has 9 elements among which are found (– 1, 0) and (0,1). Find I. The set ������ II. Remaining elements of ������ × ������. Q5. Let ������ = {1,2,3,4}, ������ = {1,5,9,11,15,16} and ������ = {(1,5), (2,9), (3,1), (4,5), (2,11)}. Mark as true/false: I. ������ is a relation from ������ to ������ II. ������ is a function from ������ to ������ Q6. If ������ = {2, 5, 7, 10} and ������ = {5, 14, 20, 35, 42}, ������ ∈ ������, ������ ∈ ������, find the set of ordered pairs (������, ������) such that 4������ − ������ > 0, ������ ∈ ������ and ������ ∈ ������. LEVEL 2 Q7. Define a relation ������ on the set ������ of natural numbers by ������ = {(������, ������): ������ = ������ + 5, ������ < 4; ������, ������ ∈ ������}. I. Write ������ in roster form II. Find the domain of ������ III. Find the range of ������ Q8. ������ = {1, 2, 3, 5} and ������ = {4, 6, 9}. Define a relation ������ from ������ to ������ by ������ = {(������, ������): |������ − ������| is odd; ������ ∈ ������, ������ ∈ ������}. I. Write ������ in roster form II. Find the domain of ������ III. Find the range of ������ Q9. For the relation ������ given by II. Find the range of ������ ������ = {(������, ������): ������ = ������ + 6 ; where ������, ������ ∈ ������ and ������ < 6} ������ I. Find the domain of ������ Q10. Let ������ be the subset of ������ × ������ defined by ������ = {(������������, ������ + ������) ∶ ������, ������ ∈ ������}. Is ������ a function from ������ to ������? Justify your answer. (Here, ������ is the set of integers) Q11. If ������(������) = ������3 − ������13, show that ������(������) + ������ (1) = 0. ������

M2 – Sets, Relations and Functions 60 M2.4 Domain and Range CONCEPTS 1. Definition and plots of functions such as modulus, GIF, signum, log ������, ������������, log ������ etc. 2. Domain and range of real functions. PRE-READING Category Book Name Chapter Name Chapter Sections 2.4.1 and 2.4.2 Compulsory NCERT Maths Class XI Relations and Functions 2 SYNOPSIS 1. FUNCTION: A relation ������ from a set ������ to set a ������ is said to be a function if every element of a set ������ has one and only one image in set ������. The function ������ from ������ to ������ is denoted by ������: ������ → ������ and ������: ������ → ������. 2. DOMAIN, CO-DOMAIN AND RANGE OF A FUNCTION : In a function ������: ������ → ������, ������ is the domain and ������ is the co-domain of ������. The range of the function is the set of images. 3. REAL VALUED FUNCTION : A function which has either ������ or one of its subsets as its range, is called a real valued function. Further, if its domain is also either ������ or a subset of ������, then it is called a real function. 4. SOME FUNCTIONS AND THEIR GRAPHS: I. Identity Function Let ������ be the set of real numbers. A real valued function ������ is defined as ������: ������ → ������ by ������ = ������(������) = ������ for each value of ������ ∈ ������. Such a function is called the identity function. M2.4

M2 – Sets, Relations and Functions 61 II. Modulus function The function ������: ������ → ������ defined by ������(������) = |������| for each ������ ∈ ������ is called modulus function or absolute valued function. For each non-negative value of ������, ������(������) is equal to ������. But for negative value of ������, the value of ������(������) is positive, i.e., ������(������) = {−������������,, ������ < 0 ������ ≥ 0 Example: If ������ = −������, then |������| = −(−������) = ������ III. Greatest integer function The function ������: ������ → ������ defined by ������(������) = [������], ������ ∈ ������ assumes the value of the greatest integer, less than or equal to ������, such a function is called the greatest integer function. From the definition of [������], we have [������] = −1, for −1 ≤ ������ < 0 = 0, for 0 ≤ ������ < 1 = 1, for 1 ≤ ������ < 2 = 2, for 2 ≤ ������ < 3 IV. Signum function Let ������ be the set of real numbers, then the function ������: ������ → ������ defined by ������(������) = |������| = 1, ������ > 0 ������ { 0, ������ = 0 is known as Signum function. −1, ������ < 0

M2 – Sets, Relations and Functions 62 V. Algebra of functions For functions ������: ������ → ������ and ������: ������ → ������, we have a. (������ + ������)(������) = ������(������) + ������(������), ������ ∈ ������ b. (������ − ������)(������) = ������(������) − ������(������), ������ ∈ ������ c. (������. ������)(������) = ������(������). ������(������), ������ ∈ ������ d. (������������)(������) = ������������(������), ������ ∈ ������, where ������ is a real number. e. (������) (������) = ������(������) , ������ ∈ ������, ������(������) ≠ 0. ������ ������(������) PRE-READING EXERCISE Q1. If ������ and ������ are real functions defined by ������ (������) = ������2 + 7 and ������ (������) = 3������ + 5, find each of the following: I. ������(3) + ������(−5) II. ������ (1) × ������(1) III. ������(−2) + ������(−1) 2 V. ������(������)−������(5), if ������ ≠ 5 IV. ������(������) − ������(−2) ������−5 Q2. Let ������(������) = ������2 + 2 and ������(������) = |������| − 1 be two real functions. Find I. (������ + ������)(1) II. (������ − ������)(1) III. (������������)(1) IV. (������) (1) V. 5������(1) ������ Q3. Let ������ = {(2, 4), (5, 6), (8, −1), (10, −3)}, ������ = {(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)} be two real functions. Then match the following I. ������ − ������ i) {(2, 4) , (8, −1) , (10, −3)} 54 13 II. ������/������ III. ������. ������ ii) {(2, 20), (8, −4), (10, −39)} iii) {(2, −1), (8, −5), (10, −16)} IN CLASS EXERCISE IN CLASS EXERCISE 1 Find the domain of the following functions Q1. ������(������) = 3������ 2������−8 Q2. ������(������) = log������ 5 Q3. ������(������) = √������ ln(2−������) Q4. ������(������) = √1 − |������| IN CLASS EXERCISE 2 Find the range of the following functions Q5. ������(������) = |������| + 2 Q6. ������(������) = |������ − 3| M2.4

M2 – Sets, Relations and Functions 63 Q7. ������(������) = 10 − 4������������ Q8. ������(������) = ������2 1+������2 Q9. ������(������) = ������������������(������������) HOMEWORK LEVEL 1 Q1. Find the domain of the function ������(������) = ������3 − 5������ + 1 Q2. Find the range of the function given by ������(������) = √������ + 3 Q3. Find the domain and range of the real function ������ defined by ������(������) = 4−������ ������−4 LEVEL 2 Q4. Find the domain and range of the function ������(������) = 1 3������−2 Q5. Find the domain and range of the function ������(������) = 1 + 1 ������ Q6. Find the domain of the function ������ given by ������(������) = √������ − 1 + 1 √6−������ Q7. Find the range of the function ������(������) = [������] + 3 where [. ] denotes the greatest integer function. Q8. Find the domain of the function ������(������) = 1 + log(2 − ������) + √������ − 2 ������−2 Q9. Find the domain of the function ������ given by ������(������) = √������+1 √5−2������ LEVEL 3 Q10. Find the domain of the function ������(������) = log(|������| − 2)

M2 – Sets, Relations and Functions 64 Test Practice Problems Purpose: To practice a mixed bag of questions in a speed based format similar to what you will face in entrance examinations. In most entrance examinations, you will get not more than 3 minutes to attempt a question. Hence you need to be able to attempt a question in less than 3 minutes, and at the end of 3 minutes skip the question and move to the next one. Approach:  Attempt the Test Practice Problems only when you have the stipulated time available at a stretch.  Start a timer and attempt the section as a test.  DO NOT look at the answer key / solutions after each question.  DO NOT guess a question if you do not know it. Competitive examinations have negative marking.  Solve as much as possible within the stipulated time, and then fill the OMR provided at the end of the TPP.  Fill the table at the end of the TPP and evaluate the number of attempts, and accuracy of attempts, which will help you evaluate your preparedness level for the chapter. TEST PRACTICE PROBLEMS – 1 No. of questions: 18 Total time: 54 Minutes Time per question: 3 Minutes Q1. If ������ is the set of letters needed to spell MATHEMATICS and ������ is the set of letters needed to spell STATISTICS, then which of the following is correct? A) ������ ⊂ ������ B) ������ − ������ = ������ C) ������ ∪ ������ = ������ − ������ D) None of these Q2. Let ������ = {������: ������ is a prime factor of 240} and ������ = {������: ������ is the sum of any two prime factors of 240}. Then, which of the following is true? A) 5 ∉ ������ ∩ ������ B) 7 ∈ ������ ∩ ������ C) 8 ∈ ������ ∩ ������ D) 8 ∈ ������ ∪ ������ Q3. If ������ and ������ both contain same number of elements and are finite sets, then A) ������(������ ∪ ������) = ������(������ ∩ ������) B) ������(������ − ������) = ������(������ − ������) C) ������(������ − ������) = ������(������) D) ������(������ − ������) = ������(������) Q4. Let ������(������) = ������ and ������(������) = ������. Then the total number of non-empty relations that can be defined from ������ to ������ is A) ������������ B) ������������ − 1 C) ������������ − 1 D) 2������������ − 1 Q5. A, B, C are the sets of letters needed to spell the words STUDENT, PROGRESS and CONGRUENT, respectively. Then ������(������ ∪ (������ ∩ ������)) is equal to. A) 8 B) 9 C) 10 D) 11 Q6. A,B,C are three sets such that ������(������) = 25, ������(������) = 20, ������(������) = 27, ������(������ ∩ ������) = 5, ������(������ ∩ ������) = 7 and ������ ∩ ������ = ∅ then ������(������ ∪ ������ ∪ ������) is equal to A) 60 B) 65 C) 67 D) 72 Q7. If ������, ������, ������ are three non-empty sets such that ������ ∩ ������ = ∅, ������ ∩ ������ = ∅, then A) ������ = ������ B) ������ ⊂ ������ C) ������ ⊂ ������ D) None of these T.P.P.

M2 – Sets, Relations and Functions 65 Q8. The domain of the function ������(������) = √− ������������(������−1) is B) (−∞, −2) ∪ (0, ∞) D) [1, ∞) √������2+2������ A) (−∞, 2] C) (−∞, −2) ∪ (0, 2] Q9. If ������(������) = 10, ������(������) = 6 and ������(������) = 5 for three disjoint sets A, B, C, then ������(������ ∪ ������ ∪ ������) equals A) 21 B) 11 C) 1 D) 9 Q10. If ������ = {������, ������}, then the power set of ������ is B) {������, ������, ������} D) {������, {������}, {������}{������, ������}} A) {������������, ������������} C) {������, {������}, {2������}} Q11. Given ������(������) = 1 ������������������ ������(������) = 1 then which of the following statements about the domains of ������ and ������ is √|������|−������ √������−|������| correct? A) ������������������ ������ ≠ ∅ and ������������������ ������ = ∅ B) ������������������ ������ = ∅ and ������������������ ������ ≠ ∅ C) ������ and ������ have the same domain D) ������������������ ������ = ∅ and ������������������ ������ = ∅ Q12. Which one of the following is (������ − ������) ∪ (������ − ������)? A) (������ ∪ ������) ∪ (������ − ������) B) (������ ∪ ������) ∪ (������ ∩ ������) C) (������ ∪ ������) − (������ ∩ ������) D) (������ − ������) ∩ (������ − ������) Q13. If ������ and ������ are two sets, then (������ − ������) ∪ (������ − ������) ∪ (������ ∩ ������) is equal to A) ������ ∪ ������ B) ������ ∩ ������ C) ������ − ������ D) (������ ∪ ������)′ Q14. Let ������ be the set of integers, ������ be the set of non-negative integers: ������������ be the set of non-positive integers; ������ is the set of even integers and ������ is the set of prime numbers. Then A) ������ ∩ ������������ = ∅ B) ������ − ������ = ������������ C) (������ − ������������) ∪ (������������ − ������) = ������ − {0} D) ������ ∩ ������ = ∅ Q15. If ������ = {1,2} and ������ = {0,1}, then ������ × ������ A) {(1,0), (1,1), (2,0), (2,1)} B) {(1,0), (2,1)} C) {(1,1), (1,2), (0,1), (0,2)} D) None of these Q16. If ������ = {1,2,3}, ������ = {3,4,5}, then (������ ∩ ������) × ������ ������������ A) {(1,3), (2,3), (3,3)} B) {(3,1), (3,2), (3,3)} C) {(1,3), (3,1), (3,2)} D) None of these Q17. If ������ = {������, ������, ������, ������} and ������ = {1,2,3}, then which of the following is a relation from ������ to ������ ? A) ������1 = {(������, 1), (2, ������), (������, 3)} B) ������2 = {(������, 1), (������, 3), (������, 2), (������, 3)} C) ������3 = {(1, ������), (2, ������), (3, ������)} D) ������4 = {(������, 1), (������, 2), (������, 3), (3, ������)} Q18. If ������ = {������, ������}, ������ = {������, ������}, ������ = {������, ������} , then {(������, ������), (������, ������), (������, ������), (������, ������), (������, ������), (������, ������)} is equal to A) ������ ∩ (������ ∪ ������) B) ������ ∪ (������ ∩ ������) C) ������ × (������ ∪ ������) D) ������ × (������ ∩ ������) TEST PRACTICE PROBLEMS – 2 No. of questions: 18 Total time: 54 Minutes Time per question: 3 Minutes Q19. The set of all ������ for which ������(������) = ������������������ ������−2 2 and ������(������) = 1 are both not defined is ������+2 √������2−9 A) (−2,2) B) [−2,2] C) (−3,3) D) [−3,3]

M2 – Sets, Relations and Functions 66 Q20. If ������ and ������ are two sets such that ������ ∩ ������ = ������ ∪ ������, then A) ������ ⊈ ������ B) ������ ⊈ ������ C) ������ = ������ D) None of these Q21. If ������ = {2������: ������ ∈ ������}, ������ = {3������: ������ ∈ ������} and ������ = {5������: ������ ∈ ������}, then ������ ∩ (������ ∩ ������) is equal to A) {15,30,45, … } B) {10,20,30, … } C) {30,60,90, … } D) {7,14,21, … } Q22. If a relation ������: ������ → ������, where ������ = {1,2,3} and ������ = {1, 3, 5}, is defined by ������ = {(������, ������): ������ < ������, ������ ∈ ������, ������ ∈ ������}, then A) ������ = {(1,3), (1,5), (2,3), (2,5), (3,5)} B) ������ = {(1,1), (1,5), (2,3), (3,5)} C) ������ = {(3,1), (5,1), (3,2), (5,3)} D) ������ = {(1,1), (5,1), (3,2), (5,3)} Q23. For non-empty subsets A and B A) Any subset of ������ × ������ defines a function from ������ to ������ B) Any subset of ������ × ������ defines a relation C) Any subset of ������ × ������ defines a function on ������ D) None of these Q24. Let ������ and ������ be two sets such that ������ × ������ = {(������, 1), (������, 3), (������, 3), (������, 1), (������, 2), (������, 2)}. Then, A) ������ = {1,2,3} ������������������ ������ = {������, ������} B) A = {a, b} and B = {1,2,3} C) ������ = {1,2,3} ������������������ ������ ⊂ {������, ������} D) ������ ⊂ {������, ������} ������������������ ������ ⊂ {1,2,3} Q25. Let ������ and ������ be two sets such that ������ × ������ has 6 elements. If three elements of ������ × ������ are {(1,4), (2,6), (3,6)}, then A) ������ = {1,2} ������������������ ������ = {3,4,6} B) ������ = {4,6} ������������������ ������ = {1,2,3} C) ������ = {1,2,3} ������������������ ������ = {4,6} D) ������ = {1,2,4} ������������������ ������ = {3,6} Q26. Let ������ be the set of integers. For ������, ������ ∈ ������, ������ ������ ������ if and only if |������ − ������| < 1, then A) ������ is not reflexive B) ������ is not symmetric C) ������ = {(������, ������); ������ ∈ ������} D) ������ is not an equivalence relation Q27. The domain of the function ������(������) = sin−1(������−3) is √(9−������2) A) [1,2] B) [2,3) C) [1,3) D) [1,3] Q28. A boating club consists of 82 members, each member is either a sailboat owner or a powerboat owner. If 53 members owned sailboats and 38 members owned powerboats, the number of members who own both sailboats and powerboats is A) 6 B) 7 C) 9 D) 4 Q29. If ������(������) is a polynomial satisfying ������(������). ������ (1) = ������(������) + ������ (1) and ������(3) = 28 then ������(4) is given by ������ ������ A) 63 B) 65 C) 67 D) 68 Q30. Let ������ and ������ be two sets such that ������ ∪ ������ = ������. Then, ������ ∩ ������ is equal to A) ������ B) ������ C) ������ D) None of these Q31. Let ������ and ������ be two sets. The (������ ∪ ������)′ ∪ (������′ ∩ ������) = A) ������′ B) ������ C) ������′ D) None of these Q32. If ������, ������, ������ be three sets such that ������ ∪ ������ = ������ ∪ ������ and ������ ∩ ������ = ������ ∩ ������, then A) ������ = ������ B) ������ = ������ C) ������ = ������ D) ������ = ������ = ������ T.P.P.

M2 – Sets, Relations and Functions 67 Q33. Let ������ and ������ be two sets such that ������ ∩ ������ = ������ ∩ ������ = ������ and ������ ∪ ������ = ������ ∪ ������ for some set ������. Then, A) ������ = ������ B) ������ = ������ C) ������ = ������ D) ������ ∪ ������ = ������ Q34. Let ������, ������, ������ be three sets such that ������ ∪ ������ ∪ ������ = ������, where ������ is the universal set. Then, {(������ − ������) ∪ (������ − ������) ∪ (������ − ������)}′ is equal to A) ������ ∪ ������ ∪ ������ B) ������ ∪ (������ ∩ ������) C) ������ ∩ ������ ∩ ������ D) ������ ∩ (������ ∪ ������) Q35. If ������ = {(������, ������) ∶ ������2 + ������2 ≤ 1, ������, ������ ∈ ������} and ������ = {(������, ������) ∶ ������2 + ������2 ≤ 4; ������, ������ ∈ ������}, then A) ������ − ������ = ������ B) ������ − ������ = ������ C) ������ − ������ = ������ D) ������ − ������ = ������ Q36. 20 teachers of a school either teach Mathematics or Physics. 12 of them teach Mathematics while 4 teach both the subjects. The number of teachers teaching Physics only is A) 12 B) 8 C) 16 D) None of these TEST PRACTICE PROBLEMS – 3 No. of questions: 18 Total time: 54 Minutes Time per question: 3 Minutes Q37. A market research group conducted a survey of 2000 consumers and reported that 1720 consumers liked product ������1 and 1450 consumers liked product ������2. The least number of consumers who must have liked both the products is A) 1170 B) 3170 C) 270 D) None of these Q38. A college awarded 38 medals in Football, 15 in Basketball and 20 in Cricket. If these medals went to a total of 58 students and only three of them got medals in all the three sports. The number of students who received medals in exactly two of the three sports is A) 18 B) 15 C) 9 D) 6 Q39. Three sets ������, ������, ������ are such that ������ = ������ ∩ ������ and ������ = ������ ∩ ������, then A) ������ ⊂ ������ B) ������ ⊃ ������ C) ������ = ������ D) ������ ⊂ ������′ D) ������ ∩ ������ ∩ ������ Q40. For any three sets ������, ������ and ������ the set (������ ∪ ������ ∪ ������) ∩ (������ ∩ ������′ ∩ ������′)′ ∩ ������′ is equal to A) ������ ∩ ������′ B) ������′ ∩ ������′ C) ������ ∩ ������ Q41. Find the domain of ������(������) = √1 − √1 − √1 − ������2. A) ������ ∈ [−1, 1] B) ������ ∈ [−1,1) C) ������ ∈ (−1,1] D) ������ ∈ (−1,1) D) [6, ∞) Q42. Find the range of ������(������) = ������2 − ������ − 3. A) (− 13 , ∞) B) [− 13 , 13) 4 44 C) [− 13 , ∞) D) None of these 4 Q43. Find the domain and range of ������(������) = √������2 − 3������ + 2. A) (0, ∞) B) [0, ∞) C) (0, ∞] D) None of these Q44. Find the range of the function ������(������) = 6������ + 3������ + 6−������ + 3−������ + 2. A) [6, ∞] B) [0, ∞) C) (6, ∞)

M2 – Sets, Relations and Functions 68 Q45. Find the domain and range of ������(������) = √������2 − 4������ + 6 A) [√2, ∞) B) [−√2, ∞) C) [√3, ∞] D) None of these Q46. Find the range of ������(������) = ������2−������+1 ������2+������+1 A) [− 1 , 3] B) [1 , ∞) 3 3 C) [1 , 3] D) None of these 3 Q47. Which of the following is not true? A) 0 ∈ {0, {0}} B) {0} ∈ {0, {0}} C) {0} ⊂ {0, {0}} D) 0 ⊂ {0, {0}} Q48. Which of the following is the empty set? B) {������ ∶ ������ is a real number and ������2 + 1 = 0} D) {������ ∶ ������ is a real number and ������2 = ������ + 2} A) {������ ∶ ������ is a real number and ������2 − 1 = 0} C) {������ ∶ ������ is a real number and ������2 − 9 = 0} Q49. Which of the following set is not a null set? B) ������ = {������|������ ∈ ������, ������2 is not positive} D) ������ = {������|������ ∈ ������, ������2 + 1 = 0} A) ������ = {������|������ ∈ ������, 2������ + 1 is even} C) ������ = {������|������ ∈ ������, ������ is odd and ������2 is even} Q50. If ������ = {1, 2, 3, 4, 5}, then the number of proper subsets of ������ is A) 120 B) 30 C) 31 D) 32 Q51. ������ − ������ is equal to B) ������ ∪ ������ C) ������ ∩ ������ D) ������ − (������ ∩ ������) A) ������ − ������ Q52. ������ − ������ = ������ iff B) ������ ⊂ ������ C) ������ = ������ D) ������ ∩ ������ = ������ A) ������ ⊂ ������ Q53. ������ − ������ = ������ − ������ if B) ������ ⊂ ������ C) ������ ∩ ������ = ������ D) ������ = ������ A) ������ ⊂ ������ Q54. If ������ and ������ are any two sets, then (������ ∪ ������) − (������ ∩ ������) = B) ������ − ������ D) None of these A) ������ − ������ C) (������ − ������) ∪ (������ − ������) TEST PRACTICE PROBLEMS – 4 No. of questions: 18 Total time: 54 Minutes Time per question: 3 Minutes Q55. Which of the following is not true? A) (������ ∩ ������) ⊂ ������ B) ������ ⊂ ������ ∪ ������ C) (������ − ������) ⊂ ������ D) ������ ⊂ (������ − ������) Q56. Let ������ = {������ ∈ ������, 1 ≤ ������ ≤ 8} , ������ = {1, 2, 3}, ������ = {2, 4, 6}, ������ = {1, 3, 5, 7}, then (������ ∪ ������)′ = ������ A) {5, 7, 8} B) {1, 3, 5, 6, 7, 8} C) {2, 4, 6, 8} D) {1, 3, 5, 7, 8} D) {3, 4, 5} Q57. ������ = {������|������2 − 7������ + 12 = 0}, ������ = {������|������2 − ������ − 12 = 0}, then ������ ∩ ������ = A) {3} B) {4} C) {−3, 3, 4} T.P.P.

M2 – Sets, Relations and Functions 69 Q58. ������ = {������|������2 − ������ − 12 = 0}, ������ = {������|������2 − 8������ + 15 = 0}, then ������ ∪ ������ = A) {3, 4, 5} B) {3, 4} C) {−3, 3, 4, 5} D) {−3, 4, 5} Q59. If ������ = {������, ������}, ������ = {������, ������}, ������ = {������, ������}, then {(������, ������), (������, ������), (������, ������), (������, ������), (������, ������), (������, ������)} = A) ������ ∩ (������ ∪ ������) B) ������ ∪ (������ ∩ ������) C) ������ × (������ ∪ ������) D) ������ × (������ ∩ ������) Q60. If (1, 3), (2, 5) and (3, 3) are three elements of ������ × ������ and the total number of elements in ������ × ������ , then the remaining elements of ������ × ������ are A) (1, 5); (2, 3); (3, 5) B) (5, 1); (3, 2); (5, 3) C) (1, 5); (2, 3); (5, 3) D) (1, 3); (2, 5); (3, 3) Q61. Let ������(������) = ������, then the number of all relations on ������ is A) 2������ B) 2������3 C) 2������2 D) ������2 Q62. Let ������ = {1, 2, 3}. The total number of distinct relations, that can be defined over ������ is A) 29 B) 6 C) 8 D) 9 Q63. The cartesian product ������ × ������ has 9 elements among which are found (−1, 0) and (0, 1). Then the set ������ is A) {−1, 0, 1} B) {−1, 0} C) {0,1} D) {−1,1} Q64. If ������ = {������, ������, ������, ������} and ������ = {1, 2, 3}, then which of the following is a relation from ������ to ������? A) ������1 = {(������, 1), (2, ������), (������, 3)} B) ������2 = {(������, 1), (������, 3), (������, 2), (������, 3)} C) ������3 = {(1, ������), (2, ������), (3, ������)} D) ������4 = {(������, 1), (������, 2), (������, 3), (3, ������)} Q65. If ������ is a relation from a finite set ������ having ������ elements to a finite set ������ having ������ elements, then the number of relations from ������ to ������ is A) 2������������ B) 2������������ − 1 C) 2������������ D) ������������ Q66. If ������ = {(������, ������)|������, ������ ∈ ������, ������2 + ������2 ≤ 4} is a relation in ������, then domain of ������ is A) {0, 1, 2} B) {0, −1, −2} C) {−2, −1, 0, 1, 2} D) {−2, −1, 0, 1} Q67. If ������(������) = ������2 − 2������ + 3, then the value of ������ for which ������(������) = ������(������ + 1) is A) 1/2 B) 1/3 C) 1 D) 3 Q68. If ������(������) = ������������2 + ������������ + 2 and ������(1) = 3, ������(4) = 42, then ������ and ������ respectively are A) −3, 2 B) 3, 2 C) −2, 3 D) 3, −2 Q69. If ������(������) = ������ + 1, such that [������(������)]3 = ������(������3) + ������������ (1), then ������ = ������ ������ A) 1 B) 3 C) −3 D) −1 D) 6 Q70. If for non-zero ������, ������ ⋅ ������(������) + ������ ⋅ ������ (1) = 1 − 5, where ������ ≠ ������, then ������(2) = ������+������ ������ ������ A) 3(2������+3������) B) 3(2������−3������) C) 3(3������−2������) 2(������2−������2) 2(������2−������2) 2(������2−������2) Q71. If in greatest integer function, the domain is a set of a real numbers, then range will be set of A) Real numbers B) Rational numbers C) Imaginary numbers D) Integers Q72. Domain and range of ������(������) = |������−3| are respectively ������−3 B) ������ − {3}, {1, −1} D) None of these A) ������, [−1,1] C) ������+, ������

M2 – Sets, Relations and Functions 70 DATA ANALYSIS Guide A # of questions Total problems in TPP B # Attempts Total attempts in OMR C # Correct Total questions correct D # Incorrect Out of the ones marked in OMR E # Unattempted ������ − ������ F Percentage attempts ������ ������ × 100 G Percentage Accuracy ������ ������ × 100 Question type # Correct (C) # Incorrect (I) # Unattempted (U) Easy Medium Hard Tip: To begin with, your accuracy must be high, typically > 60%. Percentage attempts should be > 50% As time progresses, your percentage attempts should increase without a reduction in accuracy. Additionally, you should be able to get > 80% easy questions correct, as they involve basic recall of the concepts and formulae of the chapter. T.P.P.

M2 – Sets, Relations and Functions 71 Answer Key PRE-TEST Natural Whole Integer Rational Irrational Real None       4    3.89    3. 3̅    −3 √−1  √2  2 3  0  ������ M2.1 INTRODUCTION TO SETS PRE-READING EXERCISE Q2. I. True II. True Q1. I. 8 ∉ ������ III. False II. 0 ∉ ������ IV. False III. 4 ∈ ������ Q3. I. 4, non-singleton set Q2. I. True II. 2, non-singleton set II. False III. 1, singleton set III. True IV. 2, non-singleton set V. 1, singleton set Q3. I. ������ ⊄ ������ VI. 3, non-singleton set II. ������ ⊂ ������ VII. 0, non-singleton set (this is a null set) III. ������ ⊂ ������ VIII. 2, non-singleton set Q4. I. {������: ������ is a student of Class XI of your school} Q4. I. [−4, 6) ⊂ {������: ������ is a student of your school} II. [−4, 6] III. (−4, 6) II. {������: ������ is a circle } ⊄ {������: ������ is a circle of radius 1 IV. (−4, ∞) cm} V. (−∞, 6] III. {������: ������ is a triangle} ⊄ {������: ������ is a rectangle} IN CLASS EXERCISE 2 IV. { ������: ������ is an equilateral triangle} ⊂ { ������: ������ is a Q5. Q. No. Tabular Form Set Builder Form triangle} V. {������: ������ is an even natural number} ⊂ {������: ������ is an I. {������, ������, ������, ������, ������} {������: ������ is a vowel of English alphabet} integer} II. {2, 4, 6, 8, 10, . . . } {������: ������ = 2������ and ������ IN CLASS EXERCISE 1 ∈ ������} Q1. I. Well-defined II. Well-defined III. Not well-defined III. {… − 3, −2, −1} {������: ������ is an integer and ������ < 0}

M2 – Sets, Relations and Functions Q6. I. ������(������) = 6 72 II. ������(������) = 499 Q6. I. Finite Set III. ������(������) = 3 Ans. II. Empty Set, Finite Set IV. ������(������) = 24 III. Singleton Set, Finite Set V. ������(������) = 9 IV. Infinite Set VI. ������(������) = 5 Q7. I. True Q7. I. Singleton set, finite set II. False II. Finite set III. True III. Empty set, finite set IV. False IV. Finite set V. False V. Empty set, finite set VI. True VI. Infinite set VII. True VII. Singleton set, finite set VIII. Infinite set Q8. I. No II. No Q8. I. {������: ������ ∈ ������, ������ ≥ −2} III. Yes II. {������: ������ ∈ ������, ������ < 10} IV. No III. {������: ������ ∈ ������} V. Yes Q9. I. (−2, 1) Q9. I. ������(������) = II. [−2, 1) {������, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}} III. (−2, 1] IV. (−∞, −2) II. 8 V. [1, ∞) III. False IV. True LEVEL 2 V. False Q10. I. No HOMEWORK II. No III. Yes LEVEL 1 Q11. I. Correct Q1. I. Not well-defined. II. Correct II. Well-defined. III. Incorrect III. Not well-defined. IV. Correct IV. Well-defined. V. Not well-defined. Q12. I. True II. True Q2. I. Equal III. False II. Equal IV. True III. Not equal IV. Not equal Q13. I. False II. True Q3. I. False III. False II. True IV. False III. False V. True Q4. I. {2, 3} Q14. I. False II. {1, 3, 5, 7, … } II. False III. {February} III. True IV. False Q5. I. {������: ������ = 5������ where ������ ∈ ������} II. {������: ������ = −������ where ������ ∈ ������} III. {������: ������ ∈ ������}

M2 – Sets, Relations and Functions 73 Q15. I. Yes Q17. I. 8 II. Yes II. 32 III. No III. 256 IV. No Q18. P(A) = {ϕ, {car}, {bike}, {bicycle}, {car, bike}, {car, Q16. I. A = {5, 6, 10, 12} bicycle}, {bike, bicycle}, {car, bike, bicycle}} II. B = {5, 7, 11} LEVEL 3 Q19. ������ = 6, ������ = 3 M2.2 SET OPERATIONS AND VENN DIAGRAMS PRE-READING EXERCISE II. III. Q1. A) ������ ∪ ������ = {1, 2, 3, 4} IV. B) ������ ∩ ������ = {2, 3} C) ������′ = {4, 5, 6, 7, 8, 9, 10} D) ������ − ������ = {1} E) ������′ = {1, 2, 6, 7, 8, 9, 10} F) ������ ∩ ������ = {3, 4} G) ������ ∪ ������′ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} H) ������ − ������ = {4} I) ������ ∩ ������ ∩ ������ = {3} J) ������ ∪ ������ ∪ ������ = {1, 2, 3, 4, 5} IN CLASS EXERCISE 1 Q1. I. ������ ∩ ������ = {������} Q4. 65 II. ������ − ������ = {������, ������, ������, ������} Q5. Proof III. ������’ = {������, ������, ������, ������, ������, ℎ, ������, ������, ������, ������, ������, ������, ������, Q6. I. 10 ������, ������, ������, ������, ������, ������, ������, ������} II. 18 Q2. A) ������ ∪ ������ = ������ III. 22 B) ������ ∪ ������ = ������ IV. True C) ������ ∪ ������ = ������ Q7. A D) ������ ∩ ������ = ������ E) ������ ∩ ������ = ������ IN CLASS EXERCISE 3 F) ������ ∩ ������ = ������ G) ������ ∪ ������′ = ������ Q8. 52 H) ������ ∩ ������′ = ������ I) (������′)′ = ������ HOMEWORK J) ������′ = ������ K) ������′ = ������ LEVEL 1 IN CLASS EXERCISE 2 Q1. I. ������ − ������ = {4, 8, 16, 20} II. ������ − ������ = {3, 6, 9, 12, 18, 21} Q3. I. III. ������ − ������ = {2, 4, 8, 10, 14, 16}

M2 – Sets, Relations and Functions 74 Q2. I. {������: ������ ������s an even natural number} Q11. ������ = {������, ������}, ������ = {������, ������} and ������ = {������, ������} II. {������: ������ is an odd natural number} III. {������: ������ ∈ ������ and ������ is not a multiple of 3} LEVEL 3 = {1, 2, 4, 5, 7, 8, … } Q12. IV. {������: ������ is a composite number or ������ = 1} = {1, 4, 6, 8, 9, 10, 12, 14, 15, 16, … . } V. { ������: ������ ∈ ������ and ������ is not a multiple of 15} VI. {������: ������ is not a perfect square} VII. {������: ������ is not a perfect cube} VIII. {������: ������ is a natural number and ������ ≠ 3} IX. {������: ������ is a natural number and ������ ≠ 2} X. {������: ������ is a natural number and ������ < 7} Q3. I. ������ ∩ (������ ∪ ������) = {7, 9, 11} II. (������ ∩ ������) ∩ (������ ∪ ������) = {7, 9, 11} III. (������ ∪ ������) ∩ (������ ∪ ������) = {7, 9, 11, 15} LEVEL 2 Q13. I. 10 II. 6 Q4. A) Region iii is empty ⟹ ������ and ������ are disjoint. B) Region ii is empty ⟹ ������ ⊂ ������ Q14. I. Region 1 C) Region iv is empty ⟹ ������ ⊂ ������ II. Region 2 D) Region i is empty ⟹ ������ ∪ ������ = ������ III. Region 3 IV. Region 4 Q5. I. 20 V. Region 5 II. 4 VI. Region 6 III. 12 VII. Region 7 VIII. Region 8 Q6. 60 Q7. Proof Q8. ������% = 39 Q9. I. 200 II. 90 III. 20 IV. 140 Q10. A M2.3 RELATIONS AND FUNCTIONS PRE-READING EXERCISE Q2. I. True II. True Q1. I. ������ × ������ = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), III. True (2, 2), (2, 3), (2, 4)} IV. True ������ × ������ = {(1, 5), (1, 6), (2, 5), (2, 6)} V. True ������ × ������ = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), VI. False (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)} Q3. I. 5 II. Yes II. 4 III. {2, 5, 7} Ans.

M2 – Sets, Relations and Functions 75 IN CLASS EXERCISE 1 Q2. I. No Q1. I. ������ = 3, ������ = 5 II. {(I, Ram), (I, Shyam), (I, Mohan), (I, Sohan), (I, Sita), (I, Gita), (II, Ram), (II, Shyam), II. ������ = 0, ������ = 0 (II, Mohan), (II, Sohan), (II, Sita), (II, Gita), III. ������ = 2, ������ = 1 (III, Ram), (III, Shyam), (III, Mohan), Q2. ������ × ������ = {(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} (III, Sohan), (III, Sita), (III, Gita),(IV, Ram), Q3. I. ������(������ × ������) = 20 (IV, Shyam), (IV, Mohan), (IV, Sohan), II. {(2, 4), (2, 6), (2, 18), (2, 54), (4, 4), (6, 6), (IV, Sita), (IV, Gita), (V, Ram), (V, Shyam), (V, Mohan), (V, Sohan), (V, Sita), (V, Gita)} (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)} III. {(Ram,I), (Ram, II), (Ram, III), (Ram, IV), IN CLASS EXERCISE 2 (Ram, V), (Shyam,I), (Shyam, II), (Shyam, III), Q4. 64 (Shyam, IV), (Shyam, V), (Mohan,I), (Mohan, Q5. I. ������ = {(1, 3), (2, 6), (3, 9), (4, 12)} II), (Mohan, III), (Mohan, IV), (Mohan, V), (Sohan,I), (Sohan, II), (Sohan, III), (Sohan, II. IV), (Sohan, V), (Sita,I), (Sita, II), (Sita, III), (Sita, IV), (Sita, V), (Gita,I), (Gita, II), (Gita, III. {1, 2, 3, 4} III), (Gita, IV), (Gita, V)} IV. A V. {3, 6, 9, 12} IV. a) ������ × ������ Q6. I. ������ = {(2, 4), (3, 9), (5, 25), (7, 49)} b) ������ × ������ II. {2, 3, 5, 7} c) ������ × ������ III. {4, 9, 25, 49} V. a) Not a relation. IN CLASS EXERCISE 3 b) Not a relation Q7. A, B c) Is a relation Q8. I. No VI. Domain = {I, II, III, IV} , codomain = {Ram, II. Yes Shyam, Mohan, Sohan, Sita, Gita}, range = III. Yes {Ram, Shyam, Mohan, Sita, Gita} IV. Yes V. Yes Q3. I. ������ = {1, 2}. ������ = {1, 3} Q9. Proof II. A = {good, bad}. B = {morning, evening} Q10. Proof III. ������ = {������, ������, ������} and ������ = {1, 2} HOMEWORK Q4. I. ������ = {−1, 0, 1} LEVEL 1 II. (−1, −1), (−1, 1), (0, −1), (0, 0), (1, −1), Q1. I. ������ = 2 and ������ = −2 (1, 0), (1, 1) II. ������ = 8 and ������ = 2 Q5. I. True II. False Q6. {(2, 5), (5, 5), (5, 14), (7, 5), (7, 14), (7, 20), (10, 5), (10, 14), (10,20), (10, 35)} LEVEL 2 Q7. I. ������ = {(1, 6), (2, 7), (3, 8)} II. {1, 2, 3} III. {6, 7, 8} Q8. I. R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)} II. {1, 2, 3, 5} III. {4, 6, 9}

M2 – Sets, Relations and Functions 76 Q9. I. {1, 2, 3} Q10. No II. {5, 7} Q11. Proof M2.4 DOMAIN AND RANGE PRE-READING EXERCISE Q6. [0, ∞) Q7. (−∞, 10) Q1. I. 6 Q8. [0, 1) II. 58 Q9. {1} III. 13 IV. ������2 + 8 HOMEWORK V. (������2−13) LEVEL 1 Q1. ������ ������−5 Q2. [3, ∞) Q3. Domain = ������ − {4}, Q2. I. 3 II. 3 Range = {−1} III. 0 IV. 0 LEVEL 2 V. 15 Q4. Domain = ������ − {2} and Range = ������ − {0} Q3. I. ������ − ������ = {(2, −1), 3 (8, −5), (10, −16)} Q5. Domain = ������ − {0} and Range = ������ − {1} Q6. [1, 6) II. ������ = {(2, 4) , (8, −1) , (10, −3)} Q7. ������ ������ 5 4 13 Q8. ������ Q9. [−1, 5) III. ������. ������ = {(2, 20), (8, −4), (10, −39)} 2 IN CLASS EXERCISE 1 Q1. ������ − {4} Q2. (0, ∞) − {1}. Q3. [0, 2) − {1} Q4. [−1, 1] IN CLASS EXERCISE 2 LEVEL 3 Q5. [2, ∞) Q10. ������ ∈ (−∞, −2) ∪ (2, ∞) TEST PRACTICE PROBLEMS Q. No. Ans. Level Mark (C) / (I) / (U) Q. No. Ans. Level Mark (C) / (I) / (U) as appropriate as appropriate Q1. D Medium Q7. D Easy Q2. D Medium Q8. C Hard Q3. B Medium Q9. A Medium Q4. D Hard Q10. D Easy Q5. B Easy Q11. A Easy Q6. A Medium Q12. C Medium Ans.

M2 – Sets, Relations and Functions 77 Q. No. Ans. Level Mark (C) / (I) / (U) Q. No. Ans. Level Mark (C) / (I) / (U) as appropriate as appropriate Q13. A Medium Q43. B Medium Q14. C Medium Q44. D Hard Q15. A Easy Q45. A Medium Q16. B Medium Q46. C Hard Q17. D Easy Q47. D Medium Q18. C Medium Q48. B Easy Q19. B Hard Q49. B Easy Q20. C Medium Q50. C Easy Q21. C Medium Q51. D Medium Q22. A Easy Q52. D Medium Q23. B Easy Q53. D Medium Q24. B Easy Q54. C Medium Q25. C Medium Q55. D Easy Q26. C Medium Q56. A Medium Q27. B Medium Q57. B Medium Q28. C Medium Q58. C Medium Q29. B Hard Q59. C Medium Q30. B Easy Q60. A Easy Q31. A Medium Q61. C Medium Q32. B Hard Q62. A Easy Q33. A Hard Q63. A Easy Q34. C Medium Q64. B Easy Q35. C Medium Q65. A Easy Q36. B Medium Q66. C Medium Q37. A Hard Q67. A Medium Q38. C Hard Q68. D Medium Q39. C Medium Q69. B Hard Q40. A Hard Q70. B Hard Q41. A Medium Q71. D Easy Q42. C Hard Q72. B Medium


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