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IDOL Institute of Distance and Online Learning ENHANCE YOUR QUALIFICATION, ADVANCE YOUR CAREER.

B.C.A 2 All right are reserved with CU-IDOL Mathematics Course Code: BCA114 Semester: First SLM UNITS : e-Lesson No.: 1 1 www.cuidol.in Unit-1 (BCA114)

Sets 33 OBJECTIVES INTRODUCTION Student will be able to : In this unit we are going to learn about the Define Set Theory. Set Theory. Illustrate the representation and types of set. Under this unit you will also understand the representation and types of set. Describe the process of operation of sets. This Unit will also make us to understand the Illustrate sets in terms of Venn diagram. Process of operation of sets and Venn diagram. www.cuidol.in Unit-1 (BCA114) INASllTITriUgThEt aOrFeDreISsTeArNveCdE AwNitDh OCNUL-IIDNOE LLEARNING

TOPICS TO BE COVERED 4 •Set •Elements of a Set •Set Description •Standard Sets and Symbols •Types of Sets •Venn-Euler Diagrams •Operation on Set www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Set 5 Definition •A set is any well-defined collection or class of distinct objects. •By a well-defined collection, we mean that there exists a rule with the help of which we should be able to say whether any given object or entity belongs to or not to the collection under consideration. •The following are some examples of set: (i) Students who speak either Hindi or English. (ii) Rivers in India. (iii) Countries in the world. (iv) Vowels in English alphabet. (v) Set of all points in a plane. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Elements of a Set 6 •The objects in a set are called its elements or members. •The elements in the set must be distinct and distinguishable. •By ‘distinct’, we mean that no element is repeated and by ‘distinguishable’, we mean that given any object, it is either in the set or not in the set. •Generally, capital letters A, B, C, X, Y, Z, etc. are used to denote set and its elements by lower case letters a, b, c, x, y, z, etc. •The symbol ϵ (epsilon) is used to indicate, belongs to (a member or element of). If x is an element of set A, we symbolically write it as x ϵ A. •The symbol ϵ is used to indicate, does not belongs to. Thus, if x is not an element of A, we write it as x A . www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Set Description 7 •The way of describing a set and its elements vary depending on the specific set. •The most common methods of describing elements of sets are as follows: Roster Method •In this method, elements of sets are described by enumerating (listing or writing) them exhaustively within braces. For example, the elements of the Set V of all the vowels in the English alphabet can be represented as: V = {a, e, i, o, u} •Sometimes, it is not possible to list all the elements, but after knowing a few elements, we can understand as to what the other elements are. For example, A = {a, b, c, …….}, the set of letters of English Alphabets. S = {1, 9, 25, 36, ...}, the set of squares of positive odd integers. •Roster method is also known as Tabular or Enumeration method. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Set Description 8 Set Builder Method •In this method, the elements of a set are represented by stating a property (qualitative, quantitative, or both) that uniquely characterize them. •To understand this method, let us consider the sets: A = {2, 4, 6, 8, ….} B = {a, e, i, o, u} •If we try to search out some property among elements of set A and B, we find that all elements of the set A are even positive integers; all the elements of set B are the vowels of English alphabets. •Thus, we can use the letter ‘x’ to represent an arbitrary element of set together with the property of x. Therefore, sets A and B may be represented as: A = {2, 4, 6, 8, …} = {x : x is an even positive integer} B = {a, e, i, o, u, ...} = {x : x is a vowel in English alphabet} Where the colon (:) is read as: such that or where and is used interchangeably with a vertical line. The symbol {x : ….} is read ‘the set of all x such that ….’. The property follows ‘:’ help us determine the element of the set that is being described. •Set Builder method is also known as Set Selection Method. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Standard Sets and Symbols 9 •The choice of name for a set is much like the choice of an identifier name in programming. •There are few sets of numbers that we will use frequently in the text, so that we have standard symbols for them, such as: N = {1, 2, 3, 4, 5, …}, the set of natural numbers. I = {….., –3, –2, –1, 0, 1, 2,…..}, the set of integers. I+ = {1, 2, 3, ….}, the set of positive integers. Q = {x ; x =p/q, where p and q are integers and q ≠ 0}, the set of rational numbers. Q+ = the set of positive rational numbers. R = the set of real numbers. C = {x : x = a + ib; a, bϵR, i = square root –1 }, the set of complex numbers. P = the set of prime numbers. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 10 Finite and Infinite Sets • A set is finite if it contains finite numbers of different elements. For example: (i) The set of months in a year. (ii) The set of students in the classrooms. (iii) The set of days in a week. (iv) The set of cities in India. • A set which contains infinite number of elements is known as infinite set. For example: N = {1, 2, 3, 4, ….}, the set of natural numbers. I = {…., –3, –2, –1, 0, 1, 2, …}, the set of integers. A = {x : x is set of points in the Euclidean planes}. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets Singleton Set 11 •A set which contains only one element is called as singleton (or unit) set. For example: A = {x : 4 < x < 6, x is an integer} B = {5} I = {x : x^2 = 9 and x is negative integer}. Null Set •A set which contains no element is called an empty set. This set is denoted by Greek letter ϕ (phi) or { }. For example: (i) ϕ = set of all those x which are not equal to x itself. = {x : x ≠ x}. (ii) A = {x : x^2 + 4 = 0, x is real}. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 12 Equality of Sets •Two sets A and B are said to be equal if every element of A is an element of B and also every element of B is an element of A. •The equality of two sets sets A and B is denoted by A = B symbolically A = B if and only if x ϵ A ↔ x ϵB. •This is known as Axiom of extension or identity. For example, if A = {4, 3, 2, 1} and B = {1, 2, 3, 4} then A = B because both have same and equal number of elements. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 13 Equivalent Sets •If the elements of one set can be put into one-to-one correspondence with the elements of another set, then the two sets are called equivalent sets. •In other words, two sets A and B are said to be equivalent sets if and only if there exists a one-to- one correspondence between their elements. •By one-to-one correspondence, we mean that for each element in A, there exists and match with one element in B and vice versa. •The symbol , ~ or ≡ is used to denote equivalent set. Example – A = {x : x is a letter in the word BOAT} B = {x : x is a letter in the word CART} Thus, A ≡ B. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 14 Sub Sets •Let A and B be two non-empty sets. The set A is a subset of B (or A is contained in B) if and only if every element of A is an element of B. •In other words, the set A is a subset of B if x ϵ A x ϵ B. Symbolically, this relationship is written as: A  B if x ϵ A  x ϵ B. which is read as ‘A is a subset of B’ or ‘A is contained in B’. •If A  B, then B is called superset of A and we write B  A which is read as ‘B is superset of A’ or ‘B contains A’. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 15 Proper Subset •A proper subset of a set AA is a subset of AA that is not equal to AA. •In other words, if BB is a proper subset of AA, then all elements of BB are in AA but AA contains at least one element that is not in BB. For example, if A={1,3,5} A={1,3,5} then B={1,5} B={1,5} is a proper subset of AA. The set C={1,3,5} C={1,3,5} is a subset of AA, but it is not a proper subset of AA since C=AC=A. The set D={1,4} D={1,4} is not even a subset of AA, since 4 is not an element of AA. Family of Sets •If the elements of a set are set themselves, then such a set is called the family of sets. •The word ‘collection’ and ‘class’ are also used for a set of sets. For example, if A = {a, b}, then the set { ϕ , {a}, {b}, {a, b}} is the family of sets whose elements are subset of the set A. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 16 Power Set •If S is any set, then the family of all subsets of S is called the power set of S and is denoted by P(S). •Obviously, ϕ and S are both members of P(S). (i) Let A = {a}, then P(A) = {ϕ, {a}}. (ii) Let A = ϕ, then P(A) = { } or {ϕ} (iii) Let A = {a, b}, then P(A) = {ϕ, {a}, {b}, {a, b}}. (iv) Let us say Set A = {a, b, c} Number of elements: 3 www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Types of Sets 17 Universal Set •In the application of set theory, all the sets under discussion are assumed to be the subset of the fixed large set, called the universal set. •This set is usually denoted by U or E. U is superset of all the sets. (i) A set of all points in the plane (ii) All the people in the world constitute universal set in any study of human population. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Venn-Euler Diagrams 18 •A device known as Venn-Euler diagram or simply Venn diagram is a pictorial representation of sets. •In Venn diagrams, a universal set U is represented by the interior of rectangle and each subset of U is represented by circle inside the rectangle. •If the sets A and B are equal, then the same circle represents both A and B. If the sets A and B are disjoint, i.e., they have no elements in common, then circle representing A and B are drawn in such a way that they have no common area. •However, if few elements are common to both A and B, then they are represented as shown in Figure 1.1. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Operation on Set 19 Union of Sets •Let A and B be two non-empty sets. The union of A and B is the set of all elements which are either in A or in B or in both A and B. •The Union of A and B is represented by A U B. It is usually read as A union B. A U B = {x : x ϵ A or x ϵ B} Example : If A = {1,2,3,4} and B = {2,6,8} then A U B = {1,2,3,4,6,8}. Intersection of Sets •Let A and B be the two non-empty sets. The intersection of A and B is the set of all elements which are in both A and B. •Intersection of A and B is denoted by A ∩ B. A ∩ B = {x : x ϵ A and x ϵ B} •To find the intersection of two sets means finding elements common to A and B. Example : If A = {1,2,3,4,5} and B = {4,5,6,7} then A ∩ B = {4,5}. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

Operation on Set 20 Complement of a Set •Let A be any set. The complement of A is a set of elements that belong to the universal set but do not belong to A. •Thus, if U is a Universal Set, the complement of A is the set U – A and is Denoted by A′ . A′ = U – A = { x : x  A} Example : If N = {1,2,3,4,….} is the universal set and A = {1,3,5,7,…}, then A′ = N – A = {2,4,6,8,….} www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

MULTIPLE CHOICE QUESTIONS 1) Let A = {3, 4, 6} and B = {1, 2}. Then A U B = __________. 21 (a) {3, 4} (b) {1} (c) {2} (d) {None} (d) {a, b} 2) Let A = {a, b} and B = {b, c, d}. Then A ∩ B = __________. (a) {b} (b) {a} (c) {b, c} 3) A __________ is an ordered collection of objects. (a) Relation (b) Function (c) Set (d) Proposition 4) Power set of empty set has exactly _________ subset. (a) One (b) Two (c) Zero (d) Three 5) The members of the set S = {x | x is the square of an integer and x < 100} is ________________ a) {0, 2, 4, 5, 9, 58, 49, 56, 99, 12} b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81} c) {1, 4, 9, 16, 25, 36, 64, 81, 85, 99} d) {0, 1, 4, 9, 16, 25, 36, 49, 64, 121} Answers: 1. (d) 2.(a) 3.(c) 4. (a) 5.(b) www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

SUMMARY 22 Let us recapitulate the important concepts discussed in this session: •A set is any well-defined collection or class of distinct objects. •The objects in a set are called its elements or members. The elements in the set must be distinct and distinguishable. •The way of describing a set and its elements vary depending on the specific set. The most common methods of describing elements of sets are - Roster Method, and Set Builder Method. •The choice of name for a set is much like the choice of an identifier name in programming. The different types of sets includes - Finite and Infinite Sets, Singleton Set, Null Set. •A device known as Venn-Euler diagram or simply Venn diagram is a pictorial representation of sets. •In Venn diagrams, a universal set U is represented by the interior of rectangle and each subset of U is represented by circle inside the rectangle. •Operations performed on sets include - Union of Sets, Intersection of Sets, Complement of a Set. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

FREQUENTLY ASKED QUESTION 23 Q:1 What is the real life applications for set? Ans: Set theory use for most when they are actually dealing with large sets of data in Python and MySQL as there are commands that are just making sets for natural join which is identified as an intersection. Programmers can use many commands to create sets and subsets. As to Python, some experts worked with the sets function in it just to identify the data and what are to be included in a subset. For Further details please refer to the SLM Unit 1. Q2: Define set. Elaborate the two most common methods of describing elements of sets. Ans: A set is any well-defined collection or class of distinct objects. The most common methods of describing elements of sets are - Roster Method, and Set Builder Method. For Further details please refer to the SLM Unit 1. Q3. What is meant by Venn diagram? Describe the following sets with example - a) Finite and Infinite Sets b) Singleton Set c) Null Set Ans: A Venn diagram is a pictorial representation of sets. In Venn diagrams, a universal set U is represented by the interior of rectangle and each subset of U is represented by circle inside the rectangle. For Further details please refer to the SLM Unit 1. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

REFERENCES 24 CU-IDOL’s “Mathematics” SLM. Seymour Lipschutz, Marc Lars Lipson, “Discrete Mathematics”, Publisher: McGraw Hill Education (India) Private Limited.  J.K. Sharma, “Discrete Mathematics”, Publisher: MacMillan India Limited. K. Chandrasekhara Rao, “Discrete Mathematics”, Publisher: Narosa Publishing House. Dr. Abhilasha S. Magar, “Applied Mathematics – I”, Publisher: Himalaya Publishing House. Dr. Abhilasha S. Magar, “Business Mathematics”, Publisher: Himalaya Publishing House. Dr. Abhilasha S. Magar, “Quantitative Methods – II”. Publisher: Himalaya Publishing House. www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

25 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-1 (BCA114) All right are reserved with CU-IDOL

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