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Basic Mathematics-I Programs Offered n e i Post Graduate Programmes (PG) l Master of Business Administration Master of Computer Applications Basic n Master of Commerce (Financial Management / Financial Technology) O Master of Arts (Journalism and Mass Communication) Mathematics-I Master of Arts (Economics) Master of Arts (Public Policy and Governance) Master of Social Work Master of Arts (English) Master of Science (Information Technology) (ODL) Master of Science (Environmental Science) (ODL) i ty Diploma Programmes Post Graduate Diploma (Management) r s e Post Graduate Diploma (Logistics) Post Graduate Diploma (Machine Learning and Artificial Intelligence) Post Graduate Diploma (Data Science) i v Undergraduate Programmes (UG) Bachelor of Business Administration Bachelor of Computer Applications Bachelor of Commerce Bachelor of Arts (Journalism and Mass Communication) U n English / Sociology) Bachelor of Social Work y Bachelor of Arts (General / Political Science / Economics / it Bachelor of Science (Information Technology) (ODL) A m c )DIRECTORATE OF Product code ( DISTANCE & ONLINE EDUCATION Amity Helpline: 1800-102-3434 (Toll-free), 0120-4614200 AMITY For Distance Learning Programmes: [email protected] | www.amity.edu/addoe DIRECTORATE OF For Online Learning programmes: [email protected] | www.amityonline.com DISTANCE & ONLINE EDUCATION (c )A m ity U ni ve r Basic Mathematics-I si ty O nl in e  e in © Amity University Press All Rights Reserved nl No parts of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. O SLM & Learning Resources Committee ty Chairman : Prof. Abhinash Kumar si Members : Dr. Divya Bansal Dr. Coral J Barboza Dr. Apurva Chauhan Dr. Monica Rose r ve Dr. Winnie Sharma Member Secretary : Ms. Rita Naskar ni U ity m )A (c Published by Amity University Press for exclusive use of Amity Directorate of Distance and Online Education, Amity University, Noida-201313  Contents e Page No. Module - I: Set Theory and Matrices 01 in 1.1 Sets 1.2 Venn diagram nl 1.3 Basic operations on Sets 1.4 DE Morgan’s laws & Distribution law 1.5 Matrix O Module – 2: Mathematical Logic 51 ty 2.1 Basic Concepts of Mathematical Logic 2.2 Switching Circuits si Module – 3: Group and Subgroup 76 3.1 Group 3.2 Sub Group and Other Groups r ve Module – 4: Graph Theory 104 4.1 Graph Theory ni Module – 5: Data Analysis 121 U 5.1 Data and their Representation 5.2 Measure of the Central Tendency 5.3 Measure of Dispersion ity 5.4 Skewness and Kurtosis m )A (c (c )A m ity U ni ve r si ty O nl in e  Basic Mathematics-I 1 Module - I: Set Theory and Matrices Notes e Course Contents: in Sets Types of sets nl Basic operations on sets Venn diagram O Cartesian products on two sets Distributive law DE Morgan’s laws ty Matrix Submatrix si Types of Matrices Addition, subtraction, multiplication of matrices Rank of matrix er Key Learning Objectives: At the end of this block, you will be able to: v 1. Define Sets ni 2. Identify types of Sets 3. Describe types of Sets operations 4. Describe Venn diagram U 5. Define DE Morgan’s laws 6. Calculate the Cartesian product of two sets 7. Compare Matrix and Submatrix ity 8. Classify Matrices 9. Describe algebra of Matrices 10. Define rank of Matrix m Structure: UNIT 1.1: Sets )A 1.1.1 Introduction 1.1.2 Definitions of Sets 1.1.3 Method of representation of Sets 1.1.3.1 Tabular method or Roster method (c 1.1.3.2 Set builder method Amity Directorate of Distance & Online Education  2 Basic Mathematics-I 1.1.4. Types of Sets Notes e 1.1.5 Subset and Superset 1.1.6 Proper subset in Unit-1.2: Venn diagram 1.2.1 Introduction nl 1.2.2 Venn diagram UNIT 1.3: Basic operations on Sets O 1.3.1 Introduction 1.3.2 Types of Sets operations 1.3.2.1 Union of Sets ty 1.3.2.2 Intersection of Sets 1.3.2.3 Disjoint of Sets 1.3.2.4 Difference of Sets si 1.3.2.5 Complement of a Set 1.3.3 Cartesian product of two sets er Unit-1.4: DE Morgan’s laws and Distribution law 1.4.1 Introduction v 1.4.2 DE Morgan’s laws 1.4.3 Distributive laws ni UNIT 1.5: Matrix 1.5.1 Introduction U 1.5.2 Matrix 1.5.3 Types of Matrices 1.5.3.1 Row and Column Matrices ity 1.5.3.2 Zero or Null Matrix 1.5.3.3 Square Matrix 1.5.3.4 Diagonal Matrix m 1.5.3.5 Rectangular Matrix 1.5.3.6 Scaler Matrix 1.5.3.7 Symmetric Matrix )A 1.5.3.8 Skew-Symmetric Matrix 1.5.3.9 Unit or Identity Matrix 1.5.3.10 Upper Triangular Matrix (c 1.5.3.1 Lower Triangular Matrix 1.5.3.12 Singular Matrix Amity Directorate of Distance & Online Education  Basic Mathematics-I 3 1.5.3.13 Non-singular Matrix Notes e 1.5.3.14 Equal Matrix UNIT 1.6: Algebra of a Matrices and Rank of Matrix in 1.6.1 Introduction 1.6.2 Algebra of a Matrices nl 1.6.2.1 Addition of Matrices 1.6.2.2 Difference of two Matrices O 1.6.2.3 Multiplication of Matrices 1.6.2.4 Negative Matrix 1.6.3 Properties of Matrix Addition ty 1.6.4 Properties of Multiplication of Matrices 1.6.5 Transpose of a Matrix 1.6.6 Properties of Transpose of Matrices si 1.6.7 Rank of Matrix v er ni U ity m )A (c Amity Directorate of Distance & Online Education  4 Basic Mathematics-I Unit - 1.1: Definition of Sets Notes e Unit Outcome: in At the end of this unit, you will learn to: Define Sets nl Demonstrate the methods of Set representation Compare various types of sets O Define subset, superset and proper subset 1.1.1 Introduction The set theory was developed by a German Mathematician Georg Cantor (1845- ty 1918). Nowadays, set theory is used in almost all branches of mathematics. We also use sets to define relations and functions. The knowledge of sets is required in the study of geometry, sequence, probability, etc. In this unit, we will discuss some basic si definitions related to sets. 1.1.2 Definition of Sets er “A well-defined collection of objects is known as a set”. Well-defined means in a given set, it must be possible to decide whether the v objects belong to the set, and by distinct, it implies that the object should not be repeated. Each object of a set is called a member or element of that set. A set is ni represented by { }. Generally, sets are denoted by capital letters X, Y, Z, etc. and its elements are denoted by small letters x, y, z, etc. U Let X be a non-empty set. If x is an element of X, then we write, and it can be read as ‘x is an element of X’ or ‘x belongs to X’. If x is not an element of X, then we write and read as ‘x is not an element of X’ or ‘x does not belong to X’. ity Example 1.1.1 Suppose we have a set X that is defined in this way X = Set of all days in a week. In this set, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday are m members of the set. 1.1.3 Method of representation of Sets )A Sets can be represented by the following two methods: 1. Tabular method or Roster method 2. Set builder method (c 1.1.3.1 Tabular method or Roster method In this method, elements are listed and put within a brace {} and separated by Amity Directorate of Distance & Online Education  Basic Mathematics-I 5 commas. Notes e Example 1.1.2 Suppose we have a set X that is defined in this way X = {x : x is an even number and x < 15) in X = {2, 4, 6, 8, 10, 12, 14} nl 1.1.3.2 Set builder method In this method, instead of listing all elements of a set, we list the property or properties satisfied by the elements of a set and write it as O X = {x : P (x)} It is read as “X is the set of all elements x such that x has the property P(x).” The symbol ‘:’ stands for such that. ty Example 1.1.3 Suppose we have a set X that is defined in this way X = Set of all even number less than 15. si X = {x: x = 2n, n ∈ N and 1 ≤ n ≤ 7} Or X = {x: x is an even number less than 15} er This method is also known as Rule method. v 1.1.4 Types of Sets: ni (i) Empty (Void/Null) set: A set which has no element, is called an empty set. It is denoted by f or {}. Example 1.1.4 Let X = Set of all even prime numbers greater than 3. U Example 1.1.5 Let Y = Set of all prime numbers less than 2. (ii) Singleton set: A set which has only one elements or members, is known as ity singleton set. Example 1.1.6 Let X = {x: x is an even prime number} and Y = {a} (iii) Finite set: A set which has a finite number of element or member, is known as a finite set. m Example 1.1.7 Let X = {x: x is an even number less than 9} and Y = {1, 3, 5, 7, 11, 13, 15} )A (iv) Infinite set: A set, which has an infinite number of elements or members, is known as an infinite set. Example 1.1.8 Let X = {x: x is a natural number} and Y = {2, 4, 6, 8, 10, 12, 14……………………} (c Amity Directorate of Distance & Online Education  6 Basic Mathematics-I (v) Equivalent sets: If two finite sets X and Y have the same number of elements, Notes e then the sets are known as an equivalent set. Example 1.1.9, Let X = {2, 4, 6, 8} and Y = {1, 3, 5, 7} in (vi) Equal sets: If X and Y are two non-empty sets and each element of X is an element of set Y, and each element of set Y is an element of set X, then sets X, and Y are called equal sets. nl Example 1.1.10 Let X = {x: x = 2n} and } and Y = {2, 4, 6, 8, 10} (vii) Universal Sets: If there are some sets under consideration, then there O happens to be a set which is a superset of each one of the given sets. Such a set is known as the universal set, and it is denoted by U. Example 1.1.11 Suppose we have three sets X = {a, b}, Y = {c,d,e} Z = {f, g, h, i, ty j}. ∴ U = {a, b, c, d, e, f, g, h, i, j} is a universal set for all given sets. (viii) Power sets: If X be a non-empty set, then the collection of all possible si subsets of set X is known as power set. It is denoted by P(X). The total number of elements in a power set of X containing n elements is. er Example 1.1.12 Let X = {a, b, c} v ∴ P (X) = {f}, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}} ni 1.1.5 Subset and Superset Let X and Y be two non-empty sets. If each element of set X is an element of set Y, then set X is known as a subset of set Y. If set X is a subset of set Y, then set Y is called U the superset of X. Also, if X is a subset of Y, then it is denoted as X ⊆ Y and read as ‘X is a subset of Y. ity If x ∈ X ⇒ x ∈ Y, then X⊆Y If x ∈ X ⇒ x ∉ Y, m Then X will not be a subset of Y. Example:1.1.13 If X = {a, b} and Y = {a, b, c, d} )A Here, each element of X is an element of Y. Thus X ⊆ Y , i.e. X is a subset of Y and Y is a superset of X. 1.1.6 Proper subset If each element of X is in set Y, but set Y has at least one element which is not in (c X, then set X is known as a proper subset of set Y. If X is a proper subset of Y, then it is written as X ⊆ Y and read as X is a proper subset of Y. Amity Directorate of Distance & Online Education  Basic Mathematics-I 7 Example 1.1.14 Notes e If N = {1, 2, 3, 4,.......} and W = {0, 1, 2, 3, 4,.....} in then N ⊆W nl Summary: A set is a well-defined collection of distinguished objects. O Set can be represented in two ways (i) Tabular form or Roster form (ii) Set builder method. In the tabular form, the elements of a set are actually written down, separated by commas and enclosed within braces. ty In the set builder method, a set is described by a characterising property of its element. si A set that does not contain a single element or member is called a null or empty set. A set which has only one element or member, is known as singleton set. er A set, which has a finite number of element or member, is known as a finite set. Otherwise, it is called a non-finite set. Two sets X and Y are said to be equal if every element of set X is in set Y and v every element of set Y is in set X. The collection of all subsets of a set X is called the Power set of X. ni Two sets X and Y are said to be equivalent, if the number of elements in both sets is equal. U All the sets under consideration are likely to be subsets of a sets is called the universal set. A set X is called a subset of a set Y. If every element of a set X is also an element ity of Y and also Y is called Superset of X. The Powerset of a set X is the collection of all subsets of X. Activity: m 1. List ten states of India that are large in their area. 2. Now write these states in Set builder form and Tabular form. 3. Now identify what kind of set it is? )A 4. If set X = {a, b, c} , then find the subset of set X. 5. If set X = {a, b, c}, then find the number of element in P(X). 6. If set X = {a, b, c}, then find the number of element in P[P(X)]. (c Amity Directorate of Distance & Online Education  8 Basic Mathematics-I Unit - 1.2: Venn Diagram Notes e Recall Session: in In the previous unit, you studied about: 1. The definition Sets nl 2. The methods of Set representation 3. The various types of sets 4. Definition of subset, superset and proper subset O Unit Outcome: At the end of this unit, you will learn to ty 1. Construct Venn diagram 1.2.1 Introduction si In the previous unit, we learned what subsets, supersets and proper subsets are. In this chapter, we will learn about what is a Venn diagram. With the help of the Venn diagram, we can easily solve some questions related to our everyday life. er 1.2.2. Venn diagram Most relationships between sets can be represented by diagrams called Venn v diagrams. The Venn diagram is named after the English logician John Venn. ni In these diagrams, rectangles and closed curves are usually circles. A universal set is usually represented by a rectangle and its subset by a circle. In a Venn diagram, the elements of a set are written in their particular set. As U shown in Fig.1.1.1. U X ity 1 2 4 6 7 m 3 5 )A Fig.1.1.1 Venn diagram In the fig. 1.1.1, U = {1, 2, 3, 4, 5, 6, 7} is a Universal set and X = {1, 4, 6, 7} is a subset. (c Amity Directorate of Distance & Online Education  Basic Mathematics-I 9 Summary: Notes e Venn diagrams are diagrams that show all the possible logical relationships between finite collections of sets. in Activity: 1. Make a set of all those properties of three less than 50 that are completely divisible nl by 4. 2. Now draw the Venn diagram of this obtained set. O ty si v er ni U ity m )A (c Amity Directorate of Distance & Online Education  10 Basic Mathematics-I Unit - 1.3: Basic Operations on Sets Notes e Recall Session: in In the previous unit, you studied about: Construction of Venn diagram nl Representation of set with the help of Venn diagram Unit Outcomes: O At the end of this unit, you will learn to 1. Define union of sets 2. Define intersection of sets ty 3. Define difference of sets 4. Define disjoint sets and complement of a set si 5. Define Cartesian product of two sets 1.3.1 Introduction er In the previous unit, we learned what Venn diagrams are and how to show a set with the help of a Venn diagram. As we know that if we apply an operation on a number such as a sum, difference, multiplication, and division, we get a new number. Similarly, v if we apply operations on a set such as union, intersection and complement, etc., we get a new set. In this unit, we will learn how many operations are in set theory and what ni their properties are and how they are applied to different types of sets. 1.3.2. Types of operations on Sets. U There are mainly four types of operations in set theory which are as follows: 1. Union of sets 2. Intersection of sets ity 3. Difference of sets 4. Complement of sets 1.3.2.1 Union of Sets: m Suppose X and Y are any two sets. The union of X and Y is the set containing all the components of X with all the elements of Y, and the common elements are taken only once. The symbol we use to denote union. Symbolically, we write it and read it X )A Union Y. (c Amity Directorate of Distance & Online Education taken only once. The symbol  we use to denote union. Symbolically we write it X Y and read it X Union Y. The union set of the two sets X and Y is the set that contains all the elements that are either in XBasic or inMathematics-I Y. Symbolically we can Write X  Y x : x  X or x Y . 11 The union set of the two sets X and Y is the set that contains all the elements that Notes e are either in X or in Y. Symbolically, we can write. in U Y X nl O Fig.1.1.2 Union of two sets. In the In the fig.fig. 1.1.2, 1.1.2, thethe total total areacovered area coveredby bytwo twocircles circlesXXand andYYare arerepresented represented by by XX∪ Y.. ty Y Example 1.3.1 If X = {1,3,5,7} and Y = {1,2,4,6,8}, then X ∪ Y = {1,2,3,4,5,6,7,8}. Here we will write the number 1 only once. Example 1.3.1 If X  1,3,5,7 and Y  1,2,4,6,8 , then si Some characteristics of operator ‘∪’ X 1. Y X   Y = Y ∪ X . Here ∪1,2,3,4,5,6,7,8 (Commutative law) we will write the number 1 only once. 2. (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z) (Associative law) 3. X ∪ ϕ = X (Identity law) er Some 4. characteristics of operator ‘  ’ X ∪ X = X (Idempotent law) ∪ 5.1. U X X =YU Y X (Law of union) (Commutative law) v 2. 1.3.2.2  Y  Z  XIntersection ofXSets: Y  Z (Associative law) ni 3. TheX  intersection X of a set X and Y is the set of all elements (Identity law) that are common in both X and Y. The symbol is used to denote intersection. The set of X and Y is the common 4. of X set theX all   X elements (Idempotent in both X and Y. Symbolically law)this is as X ∩ Y. we can write U U intersection 5. The X U of the two sets X and Y(Law ofSet is the union) of all the elements that are in both X and Y. Symbolically, we can write it as X ∩ Y = {x : x ∈ X and x ∈ Y}. 1.3.2.2 Intersection of Sets: The intersection of a set X and Y is the set of all elements that ity U are common in both X and Y. The symbol  is used to denote intersection. The set of X Y and Y is the common set of all the elements in both X and Y. Symbolically we can write this X is as X Y. m 11 )A Here, the common area of two circles is the intersection of set X and set Y. Example 1.3.2 If X = {1,3,5,7} and Y = {1,2,3,4,6,8} then X ∩ Y = {1,3}. Some characteristics of operator ‘∩’ (c 1. X ∩ Y = Y ∩ X (Commutative law) 2. (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z) (Associative law) Amity Directorate of Distance & Online Education  12 Basic Mathematics-I 3. X ∩ ϕ = ϕ (Identity law) Notes e 4. X ∩ X = X (Idempotent law) 5. U ∩ X = X (Law of union) in 6. X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z) (Distributive law) 1.3.2.3 Disjoint Sets: Two sets X and Y are known as disjoint sets, if X ∩ Y = ϕ, i.e., if X nl and Y have no common element. The Venn diagram of disjoint sets as shown in the figure: O U X Y ty si Example 1.3.3 If X  2,4,6 and Y  1,3,5 , then X Y  . So, X and Y are disjoint sets. Example 1.3.3 If X = {2,4,6} and Y = {1,3,5}, er 1.3.2.4then Difference of two Sets: If X and Y two non-empty sets, then difference X and Y is a X ∩ Y = ϕ. So, X and Y are disjoint sets. set of all those elements which are in X and not in Y. It is denoted as X Y. If the difference of1.3.2.4 Difference two sets is Y  of X ,two Sets: then it is a set of those elements which are in Y but not in X. v If X and Symbolically Y two it can non-empty be written as sets, then difference X and Y is a set of all those elements which are in X and not in Y. It is denoted as X – Y. If the difference of two sets ni Hence, is Y – X, thenXitis  Ya setxof: xthose  X and  x Ywhich are in Y but not in X. Symbolically it elements can be written as and Hence, Y XX  x : x Y and x  X – Y = {x : x ∈ X and x ∉ Y}.  U The Venn and diagram of Y – YX X =and {x : Y x∈ arex as X and Y shown in the figure and shaded region ∉ X}. represents and Y ofX Xin –figure X  Ydiagram The Venn Y and1.1.5 Y –and figure X are as1.1.6 shownrespectively. in the figure and shaded region represents X – Y and Y – X in figure 1.1.5 and figure 1.1.6 respectively. ity U U m X Y Y X )A Fig. 1.1.5 Difference of two sets Fig. 1.1.6 Difference of two sets Example 1.3.4 If X  1,2,3,4,5,6 and Y  1,3,5 , Example 1.3.4 If X = {1,2,3,4,5,6} and Y = {1,3,5}, (c then 2,4,6. X Y  then X – Y = {2,4,6}. Example 1.3.5 If Y  a, b, c , d , e, f  and X  a, e , Amity Directorate of Distance & Online Education then Y b, c,d , f . X  Example 1.3.4 If X  1,2,3,4,5,6 and Y  1,3,5 , then 2,4,6. X Y  Example 1.3.5 If Y  a, b, c , d , e, f  and X  a, e , Basic Mathematics-I 13 then YExample   X 1.3.5  b, c, dIf,Yf =.{a,b,c,d,e,f} and X = {a,e}, Notes e then Y – X = {b,c,d,f} 1.3.2.5 Complement of a Set: Suppose U is a universal set and X is a subset of U, then the in 1.3.2.5 Complement of a Set: complementary set of X is the set of complements of U that are not components of X. Suppose U is a universal set and X is a subset of U, then the complementary set ' c Symbolically of X is the we set represent the complement of complements of U that of areX relative to U withof not components theX.symbol X or Xwe. Symbolically, nl C represent The the complement Venn diagram of X relative of complement of a to setUXwith theshown is as symbol or figure inX′the X. and shaded portion The Venn' diagram of complement of a set X is as shown in the figure and shaded represents X. O portion represents X′. U ty X si 13 Fig.1.1.7 Complement of set X er Example 1.3.6 If U = {1,2,3,4,5,6,7,8} and X = {2,4,6,8}, then X′ = {1,3,5,7}. Some characteristics of complementary Set v 1. X ∪ X′ = U 2. X ∩ X′ = ϕ ni 3. (X′)′= X 4. ϕ′ = U 5. U′ = ϕ U 1.3.3 Cartesian product of two sets Let A and B be two non-empty sets. The cartesian product of A and B is denoted by ity and is defined as the set of all ordered pairs (a, b), where a ∈ A and b ∈ B. Symbolically, A × B = {(a,b) : a ∈ A and b ∈ B}. Example 1.3.7 Let A = {1,2,3} and B = {x,y} m ∴ A × B = {(1,x), (1,y), (2,x), (2,y), (3,x), (3,y)} ∴ B × A = {(x,1), (x,2), (x,3), (y,1), (y,2), (y,3)} )A Summary: The union of two sets X and Y is the set of all those components which is either in X or in Y. The intersection of two sets X and Y is the set of all the elements which are (c common in both X or Y. If the intersection of two sets is, it is called disjoint sets. Amity Directorate of Distance & Online Education  14 Basic Mathematics-I The difference of two sets X and Y is the set of all the elements that are in X nut Notes e not in Y. The difference of two sets Universal Set U and any set X is called the complement in set of X. Cartesian product of two sets A × B = {(a,b) : a ∈ A and b ∈ B}. nl Activity: O 1. If X is the set of all months in a year and Y is the set of all month in a year which name started with J, then find the followings: 2. Union of set X and set Y ty 3. Intersection of set X and set Y 4. X–Y 5. Y–X si 6. Complement of set X v er ni U ity m )A (c Amity Directorate of Distance & Online Education  Basic Mathematics-I 15 Unit - 1.4: DE Morgan’s law Notes e Recall Session: in In the previous unit, you studied about: The union of sets. nl The intersection of sets The disjoint sets O The difference between sets The complement of a set The cartesian product of two sets ty Unit Outcome: At the end of this unit, you will learn to si 1. Define DE Morgan’s law 2. Define Distributive law er 1.4.1 Introduction In the previous unit, we learned about the various types of operations of a set. In this unit, we will learn about what is DE-Morgan’s law. v ni 1.4.2 DE Morgan’s law According to this rule, “The complement of the union of the two sets is the intersection of their complementary sets.” And “The complement of intersection of the U two sets is the union of their complementary sets.” Symbolically, we represent it as 1. (X ∪ Y)′= X′ ∩ Y′ 2. (X ∩ Y)′= X′ ∪ Y′ ity Example. 1.4.1 Let U = {1,2,3,4,5,6}, A = {2,3} and B = {3,4,5}. Show that (A ∪ B)′ = A′ ∩ B′. Answer: m Given U = {1,2,3,4,5,6} A = {2,3} B = {3,4,5} )A A ∪ B = {2,3,4,5} (A ∪ B)′ = {1,6} Also, A′ = {1,4,5,6} (c Amity Directorate of Distance & Online Education  16 Basic Mathematics-I B′ = {1,2,6} Notes e A′ ∩ B′ = {1,6} Hence, (A ∪ B)′= A′ ∩ B′ in Example 1.4.2 If A and B are two sets, then find the value of A ∩ (A ∪ B)′. Answer: nl A ∩ (A ∪ B)′ = A ∩ (A′ ∩ B′) [by De-Morgan’s Law] = (A ∩ A′) ∩ B′ [by associative Law] O = ϕ ∩ B′ [∴ A ∩ A′ = ϕ] =ϕ ty 1.4.3 Distributive laws 1. A ∪ (B∩C) = (A∪B) ∩ (A∪C) 2. A ∩ (B∪C) = (A∩B) ∪(A∩C) si Example 1.4.4. If A and B are non-empty sets, then find the value of. Solution: er (A ∩ B) ∪ (A – B) = (A ∩ B) ∪ (A ∩ B') [∴ A – B = A ∩ B] = A ∩ (B ∩ B') [by distributive Law] v = A ∩ U (∴ B ∩ B' = U) ni = A Example 1.4.5 For any two sets A and B, prove the following: A ∩ (A′ ∪ B) = A ∩ B U Solution: From L.H.S. ity = A ∩ (A′ ∪ B) = (A ∩ A′) ∪ (A ∩ B) = ϕ ∩ (A ∩ B) [∴ A ∩ A′ = ϕ] m =A∩B Hence Proved. )A Example 1.4.6 If A = {1,2,3,6}, B = {2,4,6,8} and C = {1,3,5,7}, then find the value of a. (A ∪ B) ∩ (A ∪ C) b. (A ∩ B) ∪ (A ∩ C) (c Solution: (a) (A ∪ B) ∩ (A ∪ C) = (A ∩ B) ∪ (A ∩ C) [by distributive law] Amity Directorate of Distance & Online Education  Basic Mathematics-I 17 = {1,2,3,6} ∩ [{2,4,6,8} ∩ {1,3,5,7}] Notes e = {1,2,3,6} ∪ f = {1,2,3,6} in = A (b) (A ∪ B) ∩ (A ∪ C) = A ∩ (B ∪ C) [by distributive law] nl = {1,2,3,6} ∩ [{2,4,6,8} ∪ {1,3,5,7}] = {1,2,3,6} ∪ {1,2, 3, 4, 5, 6, 7, 8} O = {1,2,3,6} = A ty Summary: De-Morgan’s law (X ∪ Y )′ = X′ ∪ Y′ si (X ∩ Y )′ = X′ ∪ Y′ Distributive laws er A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) v Activity: a. If X and Y are two sets, then find the value of. ni U ity m )A (c Amity Directorate of Distance & Online Education  18 Basic Mathematics-I Unit - 1.5: Matrix Notes e Recall Session: in In the previous unit, you studied about: DE Morgan’s law nl Unit Outcome: At the end of this unit, you will learn to O 1. Define Matrix 2. Describe various types of Matrices 3. Describe Algebra of Matrices ty 4. Rank of Matrix 1.5.1 Introduction si In the previous unit, we learned about De Morgan’s law and in the earlier units we have learned the union, intersection and difference of the two sets. er Knowledge of matrices is required in various branches of mathematics. Matrix is one of the most powerful tools of mathematics. Compared to other straightforward methods, this mathematical tool makes our work much easier. The concept of the v matrix evolved as an attempt to solve the system of linear equations in a short and simple form. Matrix notation or representation and operations are used to create ni electronic spreadsheet programs that are used in various fields of commerce and science like budgeting, sales projection, cost estimation, analysis of the result of an experiment, etc. In this unit, we will discuss the matrix, equal matrix and its various U types. 1.5.2 Matrix An arrangement of mn numbers or functions in the form of m horizontal lines ity (called rows) and n vertical lines (called columns), is called a matrix of the type m by n (or m×n). Such an array is enclosed by the bracket [ ]. m Each number constituting the matrix is called an element of the matrix. The location of each element in the matrix is fixed. Therefore, the elements of the matrix are represented by letters which have two subscripts. The first subscript row )A and the second subscript column reveal which row and which column the element is in. Thus, the element in the ith row and jth column is written with aij. Therefore, the matrix in the m row and n column is often written as follows (c Amity Directorate of Distance & Online Education  The location of each element in the matrix is fixed. Therefore, the elements of the matrix are represented by represented letters whichbyhave letters twowhich have two subscripts. subscripts. The first Therow subscript firstand subscript row and the second the second subscript columnsubscript column reveal which rowreveal whichcolumn and which row andthe which column element is in.the element Thus, is in. Thus, the element in the element in ththe th row and jth the ith ith rowisand column jth column written with ais written with aij. Therefore, the matrix in the m row and n column ijij. Therefore, the matrix in the m row and n column Basic Mathematics-I is often written as follows 19 is often written as follows  a11 a11nn a11 a1n  Notes e 11   Ammnn   Amn        in a a  a m1 amn   mm11 mn  mn This is called This ais matrix called of This is m × n of a matrix. m anmatrix called. of m  n. nl The first The first letter of the m ×ofThe letter nthefirst m letter represents the the mthe of number n represents ofnrows represents number the number of rows in the inA, matrix the of rows matrix and the in the A, and matrix the A, and the second second second letter, the number of itsletter, columns. the number of its columns. letter, the number of its columns. Hence, we have Hence, we have Hence, we have O 1.5.3.1 Row and Column matrix- A matrix having 2 is5called 2 5only one row  2 5  a row matrix while 2 8 3  2 8only3 one column is known   3 2 matrix. 2 a83column 7  aamatrix. 8 7 a  isisaa522×3 3and  2 matrix. the one having isand a as matrix,  4  and 73matrix, matrix,   3 × 2 matrix.  ty 5 7 4       3 4   23 4  and2Column  4  isisacalled Example 1.5.3.1 1.5.1 Row 1.5.3 Types of Matrices  is a 1Amatrix 6 3matrix- 3 rowhaving matrix while only one row   31arow column matrix. matrix while si the one having only one column is known 1.5.3 Types 1.5.3 Types of Matrices  5  as a column matrix. of Matrices 1.5.3.1 Row1.5.3and Types of Matrices Column matrix- A matrix having only one row is called a row 1.5.3.1 Row and Column matrix- A matrix having only one row is called a row matrix while matrix while the one having only one column is known as a column  2 matrix.  the one having only one column is known as a column matrix.        Example 1.5.3.2 Null 2 6An 3m  1.5.1Matrix- is n 1  3 each a matrix row matrix of whose er while elements 4 isisa 0,3iscalled 1 column a nullmatrix. matrix of 19  2  19 Example m n the type Example 1.5.1. 6 3]2 is6a 3(1 [21.5.1   isisa while 4 5  a3(31×column  is×a 3) 1row matrix 3 row while matrix   1) column matrix. matrix. 0 0 0   5  v 0 0 0    whose 0 0  Example 1.5.3.2 Null1.5.2 Matrix-  An m n, matrix0 0each0of and  elements  are null is 0, matrices is called a nullof matrix the type of 0 0 0   0 0 ni 1.5.3.2 Null Matrix- 1.5.3.2 the type m  n.  An m × n matrix Null Matrix- An m  n 0 0 0  each of whose  each of whose elements matrix elements  is 0, is called a null is 0, is called a null matrix of matrix of the type. the type m  n. 2  3, 3  3 and 2  2 respectively. 00 0 0 00 0 00 00 0   and 00 0 0are  U Example 1.5.2 Example 1.5.2  , ,0 0 0 0 00 and are null matrices null matrices of the type of the type Example 1.5.2  0 0 0 0      0 0 0 0  are null matrices of   0 0   00 0 0 00    1.5.3.3 Square Matrix- An m n matrix for which m  n , i.e., the number of rows equals × 3, 23×3, 3 3and  3 2and × 22 2 respectively. the type the2number isrespectively. 2  3, 3  3 and 2  2 respectively. of columns called a square matrix of order n or an n- rowed square matrix. ity 1.5.3.3 Square Matrix- An m × n matrix for which m = n, i.e., the number of rows 1.5.3.3 Square Matrix- An m  n matrix for which m  n , i.e., the number of rows equals equals the number of columns is called a square matrix of order n or an n-rowed square An m n matrix matrix, awhich m fornornan, n-i.e., ij  A matrix i thej square 1.5.3.3 Square The elements matrix. aMatrix- the number of columns ij of a square is called a squarefor of order which rowed ,numbermatrix. of rows are called equals the diagonal  nn m the number The elements of acolumns is calledmatrix, of a square a square A =matrix [aij]n×nof for order n or ian = n-j, rowed squarethematrix. elements of theij matrix and the line along which theywhich lie is called are called or simply the diagonal of The elements diagonal elements aij of a square of the matrix and the line A matrix,  aijwhich along  forthey nn liei is which j , are called  called the diagonal or simply the theof diagonal matrix. the matrix. elements of the matrix and the line along which they lie is called or simply the diagonal of A  aij nn for which i  j , are called the diagonal )A The elements aij of a square matrix, the matrix. 1 9 25 Example1.5.3 elements 1.5.3 4 12 of the matrix 930 1 and  is a along the25line squarewhich matrix of order order 3. Example is a square matrixthey of lie is3.called or simply the diagonal of     Example 1.5.3 4 12 30 is a square matrix of order 3. the matrix. 8 16 32  8 16 32 1 9 A25 (c 1.5.3.4 Diagonal  Matrix-  matrix in which each one of the non-diagonal square Example 1.5.3  matrices is 0, is called 1.5.3.4 Diagonal 1.5.3.4 4aMatrix- 12Matrix- A30  is amatrix diagonal Matrix. square square inmatrix ofeach whicheachorder of3.the oneone ofnon-diagonal the non-diagonal matrices ismatrices is Diagonal A square  matrix in which 0, is called 8a diagonal 0, isacalled diagonal 16 32 Matrix. Matrix. Amity Directorate of Distance & Online Education  1 0 0  1 0 0   Example 1.5.4 0 2 0 is a diagonal matrix of order 3  3. Example 1.5.3.4  Matrix- 1.5.4 0 Diagonal  2 0  3is a diagonal matrix of order 3  3.  0 A0 square   matrix in which each one of the non-diagonal matrices is 0 0Matrix. 0, is called a diagonal 3  8 16 32 1.5.3.4 Diagonal Matrix- A square matrix in which each one of the non-diagonal matrices is 20 0, is called a diagonal Matrix. Basic Mathematics-I 1 0 0  Notes   is a diagonal matrix of order 3 3. e Example1.5.4 Example 1.5.4 0 2 0 is 1 2 3  a diagonal matrix of order 3 × 3.   Example 1.5.5  0 0 3 is a rectangular matrix of order 2  3. in 2 3 4  1.5.3.5 Rectangular Matrix- A m×n matrix in which the number of rows and same,1 1 22 3 3 a rectangular columns Example Example is not1.5.5 1.5.3.5 Rectangular1.5.5 is called 221 332 443isisAaam Matrix- matrix.matrix rectangular rectangular  n matrix matrix in whichof order of the 2 order 233.. of rows and columns is number nl Example 1.5.3.6 Scalar1.5.5Matrix-   is amatrix A diagonal rectangular in whichmatrix all theof order diagonal 2 elements 3. are equal is called not same, is called 32 43 matrix. 21 a rectangular aExample scalar matrix. Example 1.5.51 2 3   is 1.5.5 is aa rectangular rectangular matrix matrix of of order order2×3 2 .3. Example 1.5.5  2 3  4isa rectangular matrix of order 2  3. 21 3 2 AA4 3diagonal  O 1.5.3.6 1.5.3.6 Scalar Scalar Matrix- 1.5.5 Matrix- diagonal matrix in matrix in which which all all the the diagonal diagonal elements elements are are equal equal isis called called Example 22 30 4 0is a rectangular matrix of order 2  3. a1.5.3.6 scalar matrix. aExample scalar Scalar Matrix- matrix. 0 2 A diagonal  is a scalar matrix in which all the diagonal elements are equal is called 3 3diagonal 1.5.3.6 1.5.6 Matrix- Scalar A0 diagonal matrixmatrix of order in which all the. elements are 20 1.5.3.6 a scalar Scalar matrix. Matrix- A diagonal  matrix in which all the diagonal elements are equal is called equal 1.5.3.6 is called a scalar Scalar Matrix-202 00 2 matrix. A 00 diagonal matrix in which all the diagonal elements are equal is called ty 1.5.3.6 Scalar Matrix- A diagonal a scalar a scalar matrix. Example Example matrix. 1.5.6  1.5.6 002 220 000isis aa scalar  matrix in which all the diagonal elements are equal is called scalar matrix matrix of order 3 of order 333.. 1.5.3.7 matrix.  matrix- A a scalarSymmetric square matrix is called a symmetric matrix if for every value of i Example1.5.6 Example 1.5.62 0002 000200 220 isisaascalar scalarmatrix matrixof order 3 of order  3. 3×3. and j aij  a ji .2 0 0   Example1.5.6 Example