quantitative techniques

Quantitative Techniques

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uts, PuneFirst Edition 2013

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Index

I. Content .................................................................... II

II. List of Tables..................................................... VIII

III. Application ........................................................ 105

IV. Bibliography ...................................................... 110

V. Self Assessment Answers .................................... 113

Book at a Glance

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Contents

Chapter I ....................................................................................................................................................... 1Matrices and Determinants ......................................................................................................................... 1Aim ................................................................................................................................................................ 1Objectives ..................................................................................................................................................... 1Learning Outcome: ........................................................................................................................................ 11.1 Introduction .............................................................................................................................................. 21.2 Matrix ....................................................................................................................................................... 2 1.2.1 Matrix definition ...................................................................................................................... 2 1.2.2 Matrix Notation ....................................................................................................................... 2 1.2.3 Matrix Equality ........................................................................................................................ 31.3 Types of Matrix ........................................................................................................................................ 3 1.3.1 Row Matrix .............................................................................................................................. 3 1.3.2 Column Matrix ......................................................................................................................... 3 1.3.3 Zero/Null Matrix ...................................................................................................................... 3 1.3.4 Square Matrix .......................................................................................................................... 4 1.3.5 Diagonal Matrix ....................................................................................................................... 4 1.3.6 Unit/Identity Matrix ................................................................................................................. 4 1.3.7 Transpose of a Matrix .............................................................................................................. 41.4 Operations on Matrices ............................................................................................................................ 4 1.4.1 Addition of Matrices ................................................................................................................ 4 1.4.1.1 Properties of Matrix Addition ................................................................................... 5 1.4.2 Subtraction of Matrices ............................................................................................................ 5 1.4.3 Multiplication of Matrices ....................................................................................................... 6 1.4.3.1 Multiplication of a Matrix by a Number ................................................................... 6 1.4.3.2 Multiplication of a Matrix by another Matrix ........................................................... 6 1.4.3.3 Properties of Multiplication of Matrices ................................................................... 71.5 Determinants ............................................................................................................................................ 7 1.5.1 Calculating Value of 2 x 2 Determinant ................................................................................... 7 1.5.2 Calculating Value of 3 x 3 Determinant ................................................................................... 8 1.5.2.1 Cofactors ................................................................................................................... 8 1.5.2.2 Expansion by Minors ................................................................................................ 81.6 Inverse of a Matrix ................................................................................................................................... 9 1.6.1 Finding Inverse for a 2 x 2 Matrix ........................................................................................... 9 1.6.2 Finding Inverse for a 3 x 3 Matrix ........................................................................................... 91.7 Solving Simultaneous Equation using Determinants ..............................................................................11 1.7.1 Solving Two Simultaneous Equations ....................................................................................11 1.7.2 Solving Three simultaneous Equations .................................................................................. 121.8 Properties of Determinants .................................................................................................................... 131.9 Difference between Matrices and Determinants .................................................................................... 14Summary ..................................................................................................................................................... 15References ................................................................................................................................................... 15Recommended Reading ............................................................................................................................. 15Self Assessment ........................................................................................................................................... 16

Chapter II ................................................................................................................................................... 18Mathematical Logic ................................................................................................................................... 18Aim .............................................................................................................................................................. 18Objective ...................................................................................................................................................... 18Learning outcome ........................................................................................................................................ 182.1 Introduction ............................................................................................................................................ 192.2 Definition ............................................................................................................................................... 19 2.2.1 Statement ............................................................................................................................... 19 2.2.2 Truth Value ............................................................................................................................. 19

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2.2.3 Truth Table ............................................................................................................................. 19 2.2.4 Compound Statements ........................................................................................................... 192.3 Statement ................................................................................................................................................ 202.4 Compound Statement ............................................................................................................................. 202.5 Connectives ............................................................................................................................................ 21 2.5.1 Negation ................................................................................................................................. 21 2.5.2 Conjunction ............................................................................................................................ 22 2.5.3 Disjunction ............................................................................................................................ 23 2.5.4 Conditional or Implication .................................................................................................... 23 2.5.5 Biconditional or Biimplication .............................................................................................. 24 2.5.6 Contrapositive, Converse and Inverse ................................................................................... 252.6 Tautology ,Contradiction and Contingency ........................................................................................... 252.7 Laws of Algebra ..................................................................................................................................... 26 2.7.1 Identity Law ........................................................................................................................... 26 2.7.2 Commutative Law .................................................................................................................. 26 2.7.3 Complement Law ................................................................................................................... 26 2.7.4 Double Negation .................................................................................................................... 26 2.7.5 Associative Law ..................................................................................................................... 26 2.7.6 Distributive Law .................................................................................................................... 26 2.7.7 Absorption Law ...................................................................................................................... 26 2.7.8 Demorgans Law .................................................................................................................... 26 2.7.9 Equivalance of Contrapositive ............................................................................................... 27 2.7.10 Others ................................................................................................................................... 27Summary ..................................................................................................................................................... 28References ................................................................................................................................................... 28Recommended Reading ............................................................................................................................. 28Self Assessment ........................................................................................................................................... 29

Chapter III .................................................................................................................................................. 31Set Theory ................................................................................................................................................... 31Aim .............................................................................................................................................................. 31Objective ...................................................................................................................................................... 31Learning outcome ........................................................................................................................................ 313.1 Definition of a Set .................................................................................................................................. 323.2 Standard Sets .......................................................................................................................................... 323.3 Representation of set .............................................................................................................................. 32 3.3.1 Tabular Form/Roaster Method ............................................................................................... 32 3.3.2 Rule Method .......................................................................................................................... 32 3.3.3 Descriptive Form ................................................................................................................... 323.4 Types of Sets .......................................................................................................................................... 33 3.4.1 Finite Set ................................................................................................................................ 33 3.4.2 Empty or Null Set .................................................................................................................. 33 3.4.3 Subset ..................................................................................................................................... 33 3.4.3.1 Proper Subset .......................................................................................................... 33 3.4.3.2 Improper Subset ...................................................................................................... 33 3.4.4 Infinite Set .............................................................................................................................. 33 3.4.5 Disjoint Sets ........................................................................................................................... 34 3.4.6 Overlapping Sets .................................................................................................................... 34 3.4.7 Universal Set .......................................................................................................................... 34 3.4.8 Equal Set ................................................................................................................................ 34 3.4.9 Complement Set ..................................................................................................................... 34 3.4.10 Equivalent Set ...................................................................................................................... 343.5 Illustration of Various Sets ..................................................................................................................... 353.6 Basic Operations on Sets ....................................................................................................................... 35 3.6.1 Intersection of Two Sets ......................................................................................................... 35

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3.6.2 Union of Two Sets ................................................................................................................. 35 3.6.3 Relative Complement or Difference of Two Sets .................................................................. 35 3.6.4 Complement of a Set .............................................................................................................. 36 3.6.5 Symmetric Difference of Two Sets ........................................................................................ 363.7 Properties of Set ..................................................................................................................................... 36 3.7.1 Commutative Law .................................................................................................................. 36 3.7.2 Associative Law ..................................................................................................................... 36 3.7.3 Distributive Law .................................................................................................................... 37 3.7.4 Identity Law ........................................................................................................................... 37 3.7.5 Complement Law ................................................................................................................... 37 3.7.6 Idempotent Law ..................................................................................................................... 37 3.7.7 Bound Law ............................................................................................................................. 37 3.7.8 Absorption Law ...................................................................................................................... 37 3.7.9 Involution Law ....................................................................................................................... 37 3.7.10 De Morgans Law ................................................................................................................ 37 3.7.11More Results ......................................................................................................................... 37Summary ..................................................................................................................................................... 40References ................................................................................................................................................... 40Recommended Reading ............................................................................................................................. 40Self Assessment ........................................................................................................................................... 41

Chapter IV .................................................................................................................................................. 43Progression ................................................................................................................................................. 43Aim .............................................................................................................................................................. 43Objective ...................................................................................................................................................... 43Learning outcome ........................................................................................................................................ 434.1 Introduction ............................................................................................................................................ 444.2 Arithmetic Progression ........................................................................................................................... 444.3 Formulae for Arithmetic Progression ..................................................................................................... 45 4.3.1 The general form of an AP .................................................................................................... 45 4.3.2 The nth term of an AP ........................................................................................................ 45 4.3.3 Sum of first n terms ( ) of an AP ......................................................................................... 454.4 Arithmetic Mean .................................................................................................................................... 454.5 Geometric Progression ........................................................................................................................... 454.6 Formulae for Geometric Progression ..................................................................................................... 46 4.6.1 The general form of a GP ....................................................................................................... 46 4.6.2 The nth term Tn of a GP ........................................................................................................... 46 4.6.3 The sum of first n terms Sn of a GP........................................................................................ 464.7 Geometric Mean ..................................................................................................................................... 474.8 Harmonic Progression ............................................................................................................................ 474.9 Formulae for Harmonic Progression ...................................................................................................... 48 4.9.1 The General Form of HP ........................................................................................................ 48 4.9.2 The nth term (Tn)of a HP........................................................................................................ 484.10 Harmonic mean .................................................................................................................................... 484.11 Comparison between AP and GP ......................................................................................................... 484.12 Important Rules on Arithmetic mean(AM),Geometric Mean (GM) and Harmonic Mean(HM) ......... 49Summary ..................................................................................................................................................... 51References ................................................................................................................................................... 51Recommended Reading ............................................................................................................................. 51Self Assessment ........................................................................................................................................... 52

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Chapter V .................................................................................................................................................... 54Probability .................................................................................................................................................. 54Aim .............................................................................................................................................................. 54Objective ...................................................................................................................................................... 54Learning outcome ........................................................................................................................................ 545.1 Introduction ............................................................................................................................................ 555.2 Definitions .............................................................................................................................................. 55 5.2.1 Experiment ............................................................................................................................. 55 5.2.2 Deterministic Experiment ...................................................................................................... 55 5.2.3 Random Experiment .............................................................................................................. 55 5.2.3.1 Examples of Performing a Random Experiment .................................................... 55 5.2.3.2 Details ..................................................................................................................... 55 5.2.3.3 Sample Space : ........................................................................................................ 55 5.2.4 Elementary Event ................................................................................................................... 55 5.2.5 Impossible Event .................................................................................................................... 55 5.2.6 Events ..................................................................................................................................... 56 5.2.7 Mutually Exclusive Event ...................................................................................................... 56 5.2.8 Compatibility ......................................................................................................................... 56 5.2.9 Independent Events ................................................................................................................ 56 5.2.10 Dependent Events ................................................................................................................ 565.3 Probability .............................................................................................................................................. 56 5.3.1 Probability of Occurrence of an Event .................................................................................. 56 5.3.2 Results on Probability ............................................................................................................ 56 5.3.3 Binomial Distribution ............................................................................................................ 57 5.3.4 Geometric Theorem ............................................................................................................... 575.4 Conditional Probability .......................................................................................................................... 57 5.4.1 Conditional probability of Dependent Events ....................................................................... 57 5.4.2 Conditional probability of Independent Events ..................................................................... 575.5. Multiplication Rule ............................................................................................................................... 57 5.5.1 Independent Events ................................................................................................................ 57 5.5.2 Dependent Events .................................................................................................................. 575.6 Steps to Solve Probability ...................................................................................................................... 585.7 Bayes Theorem ...................................................................................................................................... 58Summary ..................................................................................................................................................... 61References ................................................................................................................................................... 61Recommended Reading ............................................................................................................................. 61Self Assessment ........................................................................................................................................... 62

Chapter VI .................................................................................................................................................. 64Permutations and Combinations .............................................................................................................. 64Aim .............................................................................................................................................................. 64Objective ...................................................................................................................................................... 64Learning outcome ........................................................................................................................................ 646.1 Introduction ............................................................................................................................................ 656.2 Basic Calculation Used .......................................................................................................................... 65 6.2.1 Factorial Notation ................................................................................................................. 656.3 Fundamental Principles of Counting ..................................................................................................... 65 6.3.1 Principle of Addition .............................................................................................................. 65 6.3.2 Principle of Multiplication ..................................................................................................... 656.4 Permutation ............................................................................................................................................ 66 6.4.1 Basic Forms of Permutations ................................................................................................. 67 6.4.1.1 All given Objects are Distinct ................................................................................. 67 6.4.1.2 When k cannot be Selected .................................................................................... 67 6.4.1.3 When all the given n objects are not distinct .......................................................... 67 6.4.1.4 Circular Permutation .............................................................................................. 67

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6.4.1.5 Repetition is Allowed .............................................................................................. 686.5 Combination ........................................................................................................................................... 686.6 Basic Forms of Combination ................................................................................................................. 68 6.6.1All Given Objects are distinct ................................................................................................ 68 6.6.2 When K objects cannot be selected ....................................................................................... 68 6.6.3 When k Objects are always Selected .................................................................................... 69 6.6.4 Distribution of Objects into two Groups ................................................................................ 69 6.6.5 Distribution of Similar Objects .............................................................................................. 69 6.6.6 Total possible Combination of n Distinct Objects ................................................................ 69 6.6.7 When All are not Distinct Objects ......................................................................................... 70 6.6.8 When all are Distinct but of Different Kind .......................................................................... 706.7 Special Case(Permutation and Combination Simultaneously) .............................................................. 706.8 Basic Manipulation on Permutation and Combinations ........................................................................ 70Summary ..................................................................................................................................................... 73References ................................................................................................................................................... 73Recommended Reading ............................................................................................................................. 73Self Assessment ........................................................................................................................................... 74

Chapter VII ................................................................................................................................................ 76Interpolation ............................................................................................................................................... 76Aim .............................................................................................................................................................. 76Objectives .................................................................................................................................................... 76Learning outcome ........................................................................................................................................ 767.1 Introduction ............................................................................................................................................ 777.2 Definition of Interpolation ..................................................................................................................... 777.3 Application ............................................................................................................................................. 777.4 Need and Importance of Interpolation ................................................................................................... 777.5 Methods of Interpolation ........................................................................................................................ 78 7.5.1 Graphical Method .................................................................................................................. 78 7.5.2 Newtons method of advancing differences ........................................................................... 78 7.5.3 Lagranges Method................................................................................................................. 78 7.5.4 Newton-Gauss Forward Method ............................................................................................ 78 7.5.5 Newton-Gauss Backward Method ......................................................................................... 79Summary ..................................................................................................................................................... 80References ................................................................................................................................................... 80Recommended Reading ............................................................................................................................. 80Self Assessment ........................................................................................................................................... 81

Chapter VIII ............................................................................................................................................... 83Consumer Arithmetic ................................................................................................................................ 83Aim .............................................................................................................................................................. 83Objectives .................................................................................................................................................... 83Learning outcome ........................................................................................................................................ 838.1 Introduction: Profit and Loss ................................................................................................................. 84 8.1.1 Formulae ................................................................................................................................ 848.2 Interest .................................................................................................................................................... 85 8.2.1 Terms Used ............................................................................................................................ 85 8.2.2 Simple Interest ....................................................................................................................... 85 8.2.2.1 Formulae ................................................................................................................. 85 8.2.3 Recurring Deposit .................................................................................................................. 86 8.2.3.1 Formulae ................................................................................................................. 86 8.2.4 Compound Interest ................................................................................................................. 86 8.2.4.1 Formulae ................................................................................................................. 86

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Summary ..................................................................................................................................................... 89References ................................................................................................................................................... 89Recommended Reading ............................................................................................................................. 89Self Assessment .......................................................................................................................................... 90

Chapter IX .................................................................................................................................................. 92Relations and Functions ............................................................................................................................ 92Aim .............................................................................................................................................................. 92Objectives .................................................................................................................................................... 92Learning outcome ........................................................................................................................................ 929.1 Relation .................................................................................................................................................. 939.2 Domain and Range of a Relation ........................................................................................................... 939.3 Functions ................................................................................................................................................ 93 9.3.1 Range, image, co-domain ...................................................................................................... 949.4 Break Even Analysis .............................................................................................................................. 94Summary ..................................................................................................................................................... 95References ................................................................................................................................................... 95Recommended Reading ............................................................................................................................. 95Self Assessment ........................................................................................................................................... 96

Chapter X .................................................................................................................................................. 98Statistics ...................................................................................................................................................... 98Aim .............................................................................................................................................................. 98Objectives .................................................................................................................................................... 98Learning outcome ........................................................................................................................................ 9810.1 Introduction .......................................................................................................................................... 9910.2 Definition of Statistics ......................................................................................................................... 9910.3 Scope and Applications of Statistics .................................................................................................... 9910.4 Characteristics of Statistics .................................................................................................................. 9910.5 Functions of Statistics ........................................................................................................................ 10010.6 Limitations of Statistics ..................................................................................................................... 10010.7 Classification ...................................................................................................................................... 10010.8 Objectives of Classification ............................................................................................................... 10010.9 Characteristics of Classification ........................................................................................................ 10010.10 Frequency Distribution .................................................................................................................... 101 10.10.1 Discrete or Ungrouped Frequency Distribution ............................................................... 101 10.10.2 Continuous or Grouped Frequency Distribution ............................................................. 101 10.10.3 Cumulative Frequency Distribution ................................................................................. 101Summary ................................................................................................................................................... 102References ................................................................................................................................................. 102Recommended Reading ........................................................................................................................... 102Self Assessment ......................................................................................................................................... 103

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List of Tables

Table 1.1 Differences between matrices and determinants .......................................................................... 14Table 2.1 Symbols of connectives ............................................................................................................... 21Table 2.2 Truth Table of Negation ............................................................................................................... 22Table 2.3 Truth table of conjunction ............................................................................................................ 22Table 2.4 Truth table for disjunction ............................................................................................................ 23Table 2.5 Truth table for implication ........................................................................................................... 24Table 2.6 Truth table of biimplication .......................................................................................................... 24Table 2.7 P P is a tautology .................................................................................................................... 25Table 2.8 contradiction ................................................................................................................................. 25Table 2.9 contingency .................................................................................................................................. 25Table 4.1 Comparison between AP and GP ................................................................................................. 48

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Chapter I

Matrices and Determinants

Aim

The aim of this chapter is to:

introduce the concept of matrices

elucidate the types of matrix

introduce determinant of matrix

Objectives

The objective of this chapter is to:

explicate the operations on matrices

describe the properties of determinants

explicate the properties of matrices

Learning Outcome

At the end of this chapter, you will be able to:

compare different types of matrix

identify the basic operations on matrix

understan d simultaneous linear equations using determinants


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1.1 IntroductionThestudyofmatricesanddeterminantsisofimmensesignificanceinthefieldofbusinessandeconomics.Thislesson introduces the matrix, the rectangular array and determinants at the heart of matrix algebra. Matrix algebra isusedquiteabitinadvancedstatistics,largelybecauseitprovidestwobenefits.

Compact notation for describing sets of data and sets of equations.Efficientmethodsformanipulatingsetsofdataandsolvingsetsofequations.

1.2 Matrix

1.2.1 Matrix definition

A matrix is a rectangular array of numbers arranged in rows and columns. It is a collection of real or complex numbers(usuallyreal)arrangedinafixednumberofrowsandcolumns.Itisarrangedinarectangularbrackets(either ( ) or [ ]).A set of real or complex numbers arranged in a rectangular array of m rows and n columns, of an order m x n (read as m by n) is called a matrix. The dimension or order of matrix is written as number of rows x number of columns.

A=

Example: A=

The topmost row is row 1.The leftmost column is column 1.

Here, the number of rows (m) is 2 and the number of columns (n) is 3.

So the matrix is of order 2 x 3 (2 by 3 matrix).

Matrices are used to solve problem in:ElectronicsStaticsRoboticsLinear programmingOptimisationIntersection of planesGenetics

1.2.2 Matrix Notation Statisticians use symbols to identify matrix elements and matrices.

Matrix elements: Consider the matrix below, in which matrix elements are represented entirely by symbols.

A=

Byconventionthefirstsubscriptreferstotherownumberandthesecondsubscriptreferstothecolumnnumber.Thusthefirstelementinthefirstrowisrepresentedby ,thesecondelementinthefirstrow,by and so on, until we reach the fourth element in second row which is represented by .

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Notation: The simplest way to represent a matrix symbolically is to use bold face letters A,B ,C etcThus A might refer to a 2 x 3 matrix in the below example

A =

Another approach of representing matrix A is:A= [ ] where i=1, 2 and j=1, 2, 3, 4This notation indicates A is a matrix with two rows and four columns. The actual element of the array are not displayed they are represented by the symbol .

1.2.3 Matrix EqualityTo understand matrix algebra, we need to understand matrix equality. Two matrices are equal if all three of the following conditions are met:

each matrix has same number of rowseach matrix has same number of columnscorresponding elements within each matrix are equal

Consider the three matrices given below A= B= C=

If A=B, then x=22 and y=33, as corresponding elements of equal matrices are equal. And it is clear that C is not equal to A or B, because C has more columns than A or B.

1.3 Types of MatrixThere are six types of matrices. They are as follows.

1.3.1 Row MatrixA matrix having a single row is called row matrix.

Example

A=

1.3.2 Column MatrixA matrix having a single column is called column matrix.

Example:

A=

1.3.3 Zero/Null MatrixA matrix having each and every element as zero is called a null or zero matrix.

Example

A=


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1.3.4 Square MatrixA matrix having equal number of rows and columns is called a square matrix.

Example

A=

1.3.5 Diagonal MatrixA square matrix having all elements zero except principal diagonal elements is called diagonal matrix. Principal diagonal elements can be any non-zero elements.

Matrix elements like , , etc are called principal diagonal elements.

Example

A=

1.3.6 Unit/Identity MatrixA square matrix which is a diagonal matrix having all principal diagonal elements as one (unit) is called identity matrix.

Example

A=

1.3.7 Transpose of a MatrixThetransposeofonematrixisobtainedbyusingtherowofthefirstmatrixasthecolumnofthesecondmatrix.

Example: if A= , then the transpose of A is represented by A

A=

1.4 Operations on MatricesLike ordinary algebra, matrix algebra has operations addition, subtraction and multiplication.

1.4.1 Addition of MatricesTwo matrices can be added only if they have same dimensions; that is, they must have same number of rows and columns.

Addition can be accomplished by adding corresponding elements.

For example, consider matrix A and matrix B

A= B=

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Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be added. Thus,

A+B =

A+B=

Andfinally,notethattheorderinwhichthematricesareaddeddoesnotaffectthefinalresult.A+B=B+A.

1.4.1.1 Properties of Matrix AdditionThe properties of addition of matrices are as follows:

Commutative property is true ;that is A+B=B+A Associative property is trueA+ (B+C) = (A+B) +CDistributive property is trueK (A+B) =ka+kb(A+B) k=Ak+BkExistence of additive identity element, if a matrix is added with null matrix of the same dimension then, it results in the same matrix, so the additive identity of a matrix is null matrixA+0=0+A=AExistence of additive inverse, if a matrix is added by inverse of A matrix, then the result is a null matrix, so the additive inverse of a matrix is the inverse of the matrix itself matrixA+ (-A) = (-A) +A= 0

1.4.2 Subtraction of MatricesLike addition of matrices, subtraction of matrices also follows the same conditions and procedures for subtracting two matrices. Two matrices can be subtracted only if they have same dimensions; that is, they must have same number of rows and columns.

Subtraction can be accomplished by adding corresponding elements.

For example, consider matrix A and matrix B

A= B=

Both matrices have the same number of rows and columns (3 rows and 2 columns), so they can be subtracted. Thus,

A-B=

A-B=

Andfinally,notethattheorderinwhichthematricesaresubtractedaffectsthefinalresult.A-B B-A.


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1.4.3 Multiplication of MatricesIn matrix multiplication there are two types of matrix multiplication. They are:

Multiplication of a matrix by a numberMultiplication of a matrix by another matrix

1.4.3.1 Multiplication of a Matrix by a NumberWhen a matrix is multiplied by a number, every element in the matrix should be multiplied by that same number. This operation produces a new matrix, which is called scalar multiple. This multiplication process is called as scalar multiplication.

For example, if x is 5 and matrix A is as follows,

A=

Then,

xA = 5A = 5 = = = B (say)

In the example above, every element of A is multiplied by 5 to produce the scalar multiple, B.

1.4.3.2 Multiplication of a Matrix by another MatrixThematrixproductABisdefinedonlywhenthenumberofcolumnsinAisequaltothenumberofrowsinB.Similarly,thematrixproductBAisdefinedonlywhenthenumberofcolumnsinBisequaltonumberofrowsinA.

Suppose that A is an i x j matrix and B is a j x k matrix. Then, the matrix product AB results in a matrix C which has i rows and k columns; and each element in C can be computed according to the following formula.

=

Where,

= the element in row i and column k in matrix C

= the element in row i and column j in matrix A

= the element in row j and column k in matrix B

= summation sign, which indicates that the should be summed over j

Suppose we want to compute AB, given the matrices below.

A= B=

Let AB = C.Because A has 2 rows, we know that C will also have 2 rows; and because B has 2 columns, we know that C will have 2 columns. To compute the value of every element in 2 x 2 matrix C, we use the formula

= , such that

= = 0 * 6 + 1 * 8 + 2 * 1 = 0+8+2 = 10

= = 0 * 7 + 1 * 9 + 2 * 2 = 0+9+4 = 13

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= = 3 * 6 + 4 * 8 + 5 * 1 = 18+32+5 = 55

= = 3 * 7 + 4 * 9 + 5 * 2 = 21+36+10 = 67

Therefore AB= C =

1.4.3.3 Properties of Multiplication of MatricesThe properties of multiplication of matrices are as follows:

Commutative property is not true ;that is ,even when matrix multiplication is possible in both direction the results may be different ,that is AB is not always equal to BAAssociative property is trueA (BC) = (AB) CDistributive property is trueK (AB) = (ka) (kb)(AB) k= (Ak) (Bk)Existence of multiplicative identity element, if a matrix is multiplied with identity matrix of the same dimension then, it results in the same matrix, so the multiplicative identity of a matrix is identity matrixAI=IA=AExistence of multiplicative inverse, if a matrix is multiplied by the inverse of it, then the result is a identity matrix, so the multiplicative inverse of a matrix is its inverse matrix (inverse of a matrix is discussed in 1.6)A * = * A = I

1.5 DeterminantsA determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of products.

example of a 2 x 2 determinant:

A=

1.5.1 Calculating Value of 2 x 2 DeterminantIngeneralweneedtofindthevalueof2x2determinantswithelementsa,b,canddasfollows:

= ad-cb

Herethediagonalsaremultiplied(topleft*bottomrightfirst)andthensubtracted.

ExampleFind the value of the determinant

= 4 * 3 - 2 * 1 = 12 2 =10 (answer)


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1.5.2 Calculating Value of 3 x 3 DeterminantA general representation of a 3 x 3 matrix is as follows

A=

Themethodusedforfindingthedeterminantsof3x3istheexpansionbyminors.

1.5.2.1 CofactorsThe 2 x 2 determinant is called the cofactor of for 3 x 3 matrix

The cofactors are formed from the elements that are not in the same row and not in the same column as .

Thus the elements in grey are not in the row and column of , so is the cofactor of .

Similarly for , the cofactor is

And for , the cofactor is

1.5.2.2 Expansion by MinorsThe3x3determinantvaluesareevaluatedbyexpansionbyminors.Thisinvolvesmultiplyingthefirstcolumnofthedeterminantwiththecofactorofthoseelements.Themiddleproductissubtractedandthefinalproductisadded.

= - +

example: evaluate = -2 (5) + 4 = -2[(-1) (2)-(-8) (4)] 5 [(2) (3) (-8) (-1)] + 4 [(3) (4)-(-1) (-1)] = -2(30)-5(-2) +4(11) =-60+10+44 = -6

Hereweareusingfirstcolumntoexpandit,evenifweusefirstrowtoexpand,itgivesthesameresult.

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1.6 Inverse of a MatrixSuppose A is an n x n matrix, denoted by ,thatsatisfiesthefollowingcondition

A = A = I

Where I is the identity matrixTo check whether inverse of the matrix exists:Find the determinant of the square matrix, if the determinant value is zero then the inverse of that matrix does not exist and that matrix is known as Singular matrix.

If the determinant value is not zero, then there exists an inverse for that matrix.

The matrix for which there is an inverse is called non-singular matrix or invertible matrix.

1.6.1 Finding Inverse for a 2 x 2 MatrixSuppose A is a non-singular 2 x 2 matrix .Then, the inverse of A can be computed as given below,

A= then = |A| is the determinant value of the matrix

How tofind the determinant value of 2 x 2matrix and 3 x 3matrix are discussed above in 1.5.1 and 1.5.2respectively.

Example:

Find the inverse of the 2 x 2 matrix B =

|B| = 4 (Refer 1.5.1)

= = is the inverse of the matrix

1.6.2 Finding Inverse for a 3 x 3 MatrixStepsforfindingtheinverseof3x3matrix:

Find the determinant of a 3 x 3 matrix, det(A)Find the transpose of the matrixFind the determinant of the cofactors of each element in the transpose matrix.Represent these values as a matrix of the cofactorsFind the adjoint of that resultant matrix adj(A)Substitute the required values in

= adj (A)Verify by multiplying A and ,the result should be an identity matrix of same dimension.

ExampleFind the inverse of A= Step 1:Find determinant of the 3 x 3 matrix (refer 1.5.2)det (A) = 1(0-24)-2(0-20)+3(0-5)det (A)=1


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Step 2:Find the transpose of the matrix

=

Step 3:Find the determinant of the cofactor of each element in the transpose matrix.

= = -24

= = -18

= = 5

= = -20

= = -15

= = 4

= = -5

= = -4

= = 1

Step 4:Represents these values as a matrix of the cofactors

Step 5:Find the adjoint of the matrix

adj (A)=

=

Step 6:Substitute the values in

= adj (A)

=

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Therefore =

Step 7:VerificationA = I

=

If a matrix is multiplied with its inverse, then the result should be the identity matrix of same dimension

1.7 Solving Simultaneous Equation using Determinants

1.7.1 Solving Two Simultaneous EquationsSystem of equation can be solved using determinants with cramers rule

The solution of (x, y) of the system

x + y = ----------- (1)

x + y = ----------- (2)

can be found using determinants

Solution:Here, x and y are the variables, & arethecoefficientsofthevariablexinequations1and2respectivelyand & arethecoefficientsofthevariableyinequations1and2respectively, and are the constants of equation 1 and 2 respectively.

Step 1:Solvethedeterminantofcoefficientsofvariablesanditisrepresentedby

=

Step 2:Solvethedeterminantreplacingconstantsinsteadofcoefficientofvariablexanditisrepresentedby

=

Step 3:Solvethedeterminantreplacingconstantsinsteadofcoefficientofvariableyanditisrepresentedby

=

Step 4:Obtain solution as x= and y =


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ExampleSolve the system using Cramers rule.x-3y=62x+3y=3

Solution:Here = 1; = -3; = 2; = 3; = 6; = 3;

So, x= = = = 3

y= = = = -1

So, the solution is (3,-1)

1.7.2 Solving Three simultaneous EquationsSystem of equation can be solved using determinants with cramers rule

The solution of (x, y, z) of the system x + y+ z = ----------- (1)

x + y+ z = ----------- (2)

x + y+ z = ----------- (3)

can be found using determinants

SolutionHere, x, y and z are the variables. , & arethecoefficientsofthevariablexinequations1,2and3respectively. , & arethecoefficientsofthevariableyinequations1,2and3respectively. , and are the coefficientsofthevariablezinequations1,2and3respectively. , and are the constants of equation 1,2 and 3 respectively.

Step 1:Solvethedeterminantofcoefficientsofvariablesanditisrepresentedby

=

Step 2:Solvethedeterminantreplacingconstantsinsteadofcoefficientofvariablexanditisrepresentedby

=

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Step 3:Solvethedeterminantreplacingconstantsinsteadofcoefficientofvariableyanditisrepresentedby

=

Step 4:Solvethedeterminantreplacingconstantsinsteadofcoefficientofvariablezanditisrepresentedby

=

Step 5:Obtain solution as x= ; y = ; z =

ExampleSolve the system using Cramers rule.2x+3y+z=2-x+2y+3z=1-3x-3y+z=0

Solution:Here = 2; = 3; = 1; = -1; = 2; = 3; = -3; =-3; = 1 and =2; = 1; =0

= = 2(11) +1(6)-3(7) = 7

So, x= = = 28 / 7=4

y= = = - 21/ 7 = -3

z= = = 21/ 7 = 3

So, the solution is (4,-3, 3)

1.8 Properties of DeterminantsThe properties of determinants are as follows:

The value of determinant remains unchanged if its rows and columns are interchangedIf any two rows/columns change by minus sign only ,then also the value of determinant remains unchangedIf any two rows/columns of a determinant are identical, then the value of determinant is zeroIf each element of a row/column of a determinant is multiplied by a same constant and then added to corresponding elements of some other row/column, then the value of determinant remains unchanged.If each element of a row/column of a determinant is zero, then the value of the determinant is zero.


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1.9 Difference between Matrices and DeterminantsFollowing is the difference between matrices and determinants

Features Matrices Determinants

Definition A matrix is an array of numbers arranged in rectangular brackets.

A determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of products.

Representation It is written inside brackets either ( ) or [ ].

It is written within two vertical Lines | | .

Value/Result It results in an array of number inside brackets.

It results in a single number.

Influence Scalar multiplication affects all the elements in a matrix.

Scalar multiplication only affects single row /single column.

Value Matrices contain many elements.

Determinant has a single number as a end result.

Nature Matrices may positive or negative.

Determinant value is always positive. Though it results in a negative number we consider it as positive because determinant is like distance(it cannot e negative whether it is forward or backward)

Table 1.1 Differences between matrices and determinants

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SummaryA matrix is a rectangular array of numbers arranged in rows and columns.A set of real or complex numbers arranged in a rectangular array of m rows and n columns, of an order m x n (read as m by n) is called a matrix.Two matrices can be added only if they have same dimensions.Commutative property, Associative property, Distributive property is true for matrix addition.ThematrixproductABisdefinedonlywhenthenumberofcolumnsinAisequaltothenumberofrowsinB.

Commutative property is not true for matrix multiplication.Associative property ,Distributive property are true for matrix multiplicationThere exists additive identity, multiplicative identity, additive inverse and multiplicative inverse for a matrix.A determinant is a square array of numbers.The 3 x 3 determinant values are evaluated by expansion by minors.System of equation can be solved using determinants with Cramers rule.

ReferencesDr. Kala, V. N. & Rana, R., 2009. Matrices, 1st ed., Laxmi Publication ltd.Jain, T. R. & Aggarwal, S. C., 2010. Business Mathematics and Statistics, V.K Enterprises.Matrices and determinants , [pdf] Available at: < http://www.kkuniyuk.com/M1410801Part1.pdf > [Accessed 31 August 2012].Gunawarden, J., Matrix algebras for beginning, [Online] Available at: < http://vcp.med.harvard.edu/papers/matrices-1.pdf > [Accessed 31 August 2012].2011, Matrices, [Video Online] Available at: < http://www.youtube.com/watch?v=9tFhs-D47Ik > [Accessed 31 August 2012].Hurst, W., Matrices & determinants, [Video Online] Available at: < http://www.youtube.com/watch?v=havr-W8IwKs > [Accessed 31 August 2012].

Recommended ReadingMcMahon, D., 2005. LinearAlgebraDemystified, McGraw-hill publication.Anton, H., 2010. Elementary Linear Algebra, 10th ed., FM Publications.Greub, W., 1975. Linear Algebra graduate texts in mathematics, Springer.


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Self AssessmentIf matrix A= 1.

and aij is the element of matrix A in ith row and jth column, then what is the value of a21?

3a. 4b. 2c. 5d.

It is given that P= 2. and Q = .What is the value of x+y if P=Q?3a. 5b. 6c. 8d.

A _3. ______ is a rectangular array of numbers arranged in rows and columns.Determinanta. Matrixb. Arrayc. Transposed.

Two matrices can be added only if they have __________.4. same dimensionsa. different dimensionsb. plus signc. minus signd.

When a matrix is multiplied by a number, then the process is called as _________.5. matrix multiplicationa. scalar multiplicationb. square multiplicationc. rectangular multiplicationd.

What type of matrix is A=6. ?squarea. diagonalb. nullc. identityd.

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What is the addition matrix of the following two matrices?7.

A = and B=

a.

b. c.

d.

If A = 8. , what is the value of 5A?

a.

b.

c.

d.

What is the value of determinant 9. ?6a. 8b. 7c. 10d.

A _________ is a square array of numbers. 10. matrixa. determinantb. arrayc. transposed.


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Chapter II

Mathematical Logic

Aim


introduce mathematical logic

describe operations on logic

highlight tautology and contradiction

Objective

The objectives of this chapter are to:

explicate logical connectives

elucidate laws of algebra of propositions

describe compound statement

Learning outcome


identify the use of mathematical logic

understand the complex procedures into simpler form

understand statement a nd the truth table

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2.1 IntroductionMathematical Logic is a tool for providing a precise meaning to mathematical statements.

It includes:A formal language for expressing themA concise notation to represent themA methodology for objectively reasoning about their truth or falsity

2.2 DefinitionThe part of mathematics concerned with the study of formal languages, formal reasoning, the nature of mathematical proof, provability of mathematical statements, computability, and other aspects of the foundations of mathematics.

2.2.1 StatementA statement is a declarative sentence which is either true or false but not both.

2.2.2 Truth ValueThe truth value of a proposition is true (T) if it is of true proposition and false (F) if it is false proposition.

ExampleP: The year 1973 was a leap year is a proposition readily decidable as false.

Note that the use of label P so that the overall statement is read p is the statement:The year 1973 was a leap year.

So we use P, Q, R, S, T to represent statements and these letters are called as statement variables, that is, variable replaced by statements.

ExampleDetermine whether the following sentences are statements are not.If it is a statement, determine its truth value.

The sun rises in west. False128= 26 False

Is 2 an integer? Not a statement as it is interrogative Take the book not a statement

2.2.3 Truth TableA table that gives the truth value of the compound statement in terms of its component part is called a truth table.

2.2.4 Compound StatementsA compound statement is a combination of two or more statements.

ExampleToday is Friday and it is a holiday


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2.3 StatementA statement is an assertion that can be determined to be True or False. A statement/simple statement or proposition is a declarative sentence that is either True or False but not both. A simple statement is the basic building block of the logic.

Those declarative statements will be admitted in the object language which have one and only one of two possible values called Truth ValueThe two truth values are true and false, which are denoted by T and F respectivelyOccasionally they are represented using symbols 1 and 0We do not use other kind of statements in object language such as exclamatory and interrogativeDeclarative sentences in object language are of two typesThefirsttypeincludesthosesentenceswhichareconsideredtobeprimitiveintheobjectlanguage

This will be denoted by distinct symbols selected from uppercase letters A, B... P, Q...Second type are obtained from the primitive ones by using certain symbols called connectives and certain punctuation marks such as parentheses to join primitive sentences

In any case, all declarative sentences to which it is possible to assign one and only of the two possible truth values are called statements.

The following are the statements which do not contain any connectives, these kinds of statements are called as atomic or primary primitive statement.

Canada is a country1. Moscow is the capital of spain2. This statement is false3. 1+101=1104. Close the door5. Toronto is an old city6. Man will reach mars by 20807.

The statements are discussed below

The statements 1 and 2 have truth values true or falseSentence3isnotastatementaccordingtothedefinition,becausewecannotassigntoitadefinitetruthvalue

If we assign a value true then the statement 3 is false, if assigned false then the statement 3 is trueSentence 4 is a statement; if the numbers are considered as decimal system then the statement is false. If it is considered as binary number system, then the statement is true. So the statement 4 is true.Statement 5 is not a statement as it is interrogativeStatement 6 is considered true in some part of the world and false in certain other parts of the worldThe statement 7 could not be determined ,it will be determined only in the year or earlier when man reaches mars before that date

2.4 Compound StatementA statement represented by a single statement variable (without any connective) is called a simple (or primitive) statement.

A statement represented by some combination of statement variables and connectives is called a compound statement.

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ExampleA dog or a car is an animalA dog is not an animal5


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P P

T F

F T

Table 2.2 Truth Table of Negation

ExampleP:The integer 10 is evenThen P: The integer 10 is not even

P:London is a city P:It is not case that London is a city P:London is not a city

P:I went to my class yesterday. P:I did not go to my class yesterday P:I was absent from my class yesterday.

P:I went to my class yesterday. P:I did not go to my class yesterday P:I was absent from my class yesterday

Negationiscalledconnectivesalthoughitonlymodifiesastatementoravariable.

2.5.2 ConjunctionLet P and Q be statements. The conjunction of P and Q, written P Q,is the statement formed by joining statements P and Q using the word and. The statement PQ is true if both P and Q are true; otherwise PQ is false.

The symbol is called and. Let P and Q be statements. The truth table of P Q is given below.

DefinitionIf p and q are statement variables, the conjunction of p and q is p and q, denoted pq.The compound statement pq .The compound statement p q is true when both p and q are true; otherwise, it is false.

P Q PQ

T T T

T F F

F T F

F F F

Table 2.3 Truth table of conjunction

ExampleP: 2 is an even integer,Q: 7 divide 14R:2 is an even integer and 7 divides 14.

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P:It is raining todayQ:There are 20 tables in this room.R:It is raining today and there are 20 tables in this room.

Jack and Jill went up the hill.From this statement we get two statement Jack went up the hill and Jill went up the hill.Then the given statement can be written symbolic from PQ.

2.5.3 Disjunction Let P and Q be statements. The disjunction of P and Q ,written P Q ,is statement formed by putting statements P and Q together using the word Or. The truth value of the statement PQ is T if atleast one of statements P and Q is true. The symbol is called Or, for the statement P Q is given below.

DefinitionIf P and Q are statement variables, the disjunction of P and Q is P or Q, denoted P Q.The compound statement P Q is true if atleast one of P or Q is true; it is false when both P and Q are false.

P Q P Q

T T T

T F T

F T T

F F F

Table 2.4 Truth table for disjunction

ExampleP:22+33 is an even integerQ:22+33 is an odd integer then P Q:22+33 is an even integer or 22+33 is an odd integer ORP Q :22+33 is an even integer or an odd integer The notation for inequalities involves and and or statements.Let a,b and c be particular real numbers.a b means a < b or a= ba


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OrQ if POrQ whenever P.In the implication P Q,P is called the hypothesis and Q is called the conclusion. The truth table of P Q is given below.DefinitionIf P and Q are statements,the statemenif P then Q or P implies Q,denoted PQ is called the conditional statement,or implication.

P Q P Q

T T T

T F F

F T T

F F T

Table 2.5 Truth table for implication

ExampleIf today is Sunday,then i will go for walk.Let P:Today is SundayQ:I will go for walk

Variety of terminaology:If P then Q Q if PP implies Q Q when PP only if Q Q follows from PPissufficientforQQisnecessaryforP

2.5.5 Biconditional or BiimplicationLet P and Q be two statements.Then P if and only if Q,written PQ is called the Biimplication or biconditional of the statement P and Q.The statement PQmayalsobereadasPisnecessaryandsufficientforQorQisnecessaryandsufficientforPorQifandonlyifPorQwhenandonlywhenP.WedefinethattheBiimplicationPQ is considered to be true when both P and Q have the same truth values and false otherwise. It is denoted by PQ.The truth table is given below.

P Q PQ

T T T

T F F

F T F

F F F

Table 2.6 Truth table of biimplication

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2.5.6 Contrapositive, Converse and InverseThe contrapostive of P Q is QPThe converse of P Q is QPThe inverse of P Q is P Q

2.6 Tautology ,Contradiction and ContingencyA tautology is a statement form where its truth values in all rows in the truth table are always true. A contradiction is a statement form where its truth values in all rows in the truth table are always false.

A contingency is a statement form that is neither tautology nor contradiction. Normally t is denoted to use tautology and c is used to denote a contradiction.

ExampleLet P, Q, R be statement variables. Show that the statement formP P is a tautologyP P is a contradiction(PQ) R is a contingency

a.

P P P P

F T T

T F T

Table 2.7 P P is a tautology

b.

P P P P

F T F

T F F

Table 2.8 contradiction

c.

P Q R PQ R (P Q) V R

F F F F T T

F F T F F F

F T F F T T

F T T F F F

T F F F T T

T F T F F F

T T F T T T

T T T T F T

Table 2.9 contingency


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2.7 Laws of AlgebraFollowing are the laws of algebra2.7.1 Identity LawP P PP T TP F PP P PP T PP F FTP PFP TPP TPT TPF PP P TP T PP F P

2.7.2 Commutative LawPQ Q PPQQPPQ Q PP Q = QP

2.7.3 Complement LawP P TP P FP P PP PF PPP

2.7.4 Double Negation( P) P

2.7.5 Associative LawP(QR) (PQ)RP(QR)(PQ)R

2.7.6 Distributive LawP(QR) (PQ)(PR)P(QR) (PQ)(PR)

2.7.7 Absorption LawP(PQ) PP(PQ) P

2.7.8 Demorgans Law(PQ)P Q(PQ)P Q

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2.7.9 Equivalance of ContrapositivePQQP

2.7.10 OthersPQP QPQ(PQ)(QP)


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SummaryMathematical Logic is a tool for providing a precise meaning to mathematical statements.A statement is a declarative sentence which is either true or false but not both.The truth value of a proposition is true (T) if it is of true proposition and false (F) if it is false proposition.A table that gives the truth value of the compound statement in terms of its component part is called a truth table.A compound statement is a combination of two or more statements.All declarative sentences to which it is possible to assign one and only of the two possible truth values are called statements.It is possible to construct rather complicated statements from simpler statements by using certain connecting words or expressions known as sentential connectives.The negation statement is generally formed by introducing the word not at a proper place in statement with the phrase It is not the case that and read as not P.Let P and Q be two statements. Then If P, then Q is the statement called an Implication or conditional statement, written P Q.

ReferencesFulda, J. S., 1993. Exclusive Disjunction and the Bi-conditional: An Even-Odd Relationship, Mathematics Magazine.Hallie, P. P., 1954. A Note on Logical Connectives, Mind 63.Bartlett, A., Simple Mathematical Logic, [Video Online] Available at: [Accessed 31 August 2012].Dr. Kirthivasan, K., Propositional Logic, [Video Online] Available at: [Accessed 31 August 2012].Lifschitz, V., Lecture notes on mathematical logic, [pdf] Available at: [Accessed 31 August 2012].Simpson, S., Mathematical Logic, [Online] Available at: [Accessed 31 August 2012].

Recommended ReadingDean McCullough, P., 1971. Logical Connectives for Intuitionist Propositional Logic, Journal of Symbolic Logic.Wansing, H., 2006. Logical Connectives for Constructive Modal Logic.Hallie, P. P., 1954. A Note on Logical Connectives, Mind 63.

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Self Assessment:Canada is a country .what kind of statement is it?1.

Primitive statementa. Compound statementb. Elementary statementc. Primary statementd.

A dog or a car is an animal. What kind of statement it is?2. Primitive statementa. Compound statementb. Elementary statementc. Primary statementd.

What is the symbol for negation?3. a. b. c. d.

What is the symbol for conjunction?4. a. b. c. d.

What is the symbol for disjunction?5. a. b. c. d.

What is the symbol for implication?6. a. b. c. d.

If P is true and Q is False what is the value of P 7. Q?Ta. Fb. Invalidc. no valued.


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Chapter III

Set Theory

Aim


defineaset

discuss different types of set

elucidate operations on sets

Objective

The objectives of this chapter are to:

discuss the types of sets

explicate null and universal set

explain Demorgans Law

Learning outcome


understand standard set

comprehend the concept of intersection and disjoint sets

understand operati ons on sets.


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3.1 Definition of a SetAsetisacollectionofwelldefinedobjectenclosedwithincurlybrackets,generallydenotedbyuppercaseletters.

The objects which form the set are called the elements or members of the set. The elements are denoted by lower case letters.

ExampleA= {1, 2, 3... 100} represents set of numbers from 1 to 100V= {a, e, i, o, u} represents set of vowels in English alphabetsThe following notation is used to show the set membershipx A means that x is a member of the set A.x A means that x is not a member of the set A.

3.2 Standard SetsFollowing are the standard setsN= set of natural numbers = {1, 2, 3, 4...}Z= set of all integers = {...,-3,-2,-1, 0,1,2,3...}R= set of all rational numbers = {p/q: p is integer, q0}Q=set of all real numbers

3.3 Representation of setFollowing are the method for representing sets3.3.1 Tabular Form/Roaster MethodIn the tabular form, all elements of the set are enumerated or listed.

ExampleThe set of natural numbers from 1 to 100 is given byN= {1, 2, 3, 4...100}The set of vowels is given byV= {a, e, i, o, u}

3.3.2 Rule MethodUnderthismethod,thedefiningpropertyofthesetisspecified.IfalltheelementsinthesethaveaPropertyP,thenwecandefinethesetasA={x: x has the property P}

ExampleN={x: 1x100, x is a natural number} or {x/1x100,xN}Where, N represents natural numberV={x/x is a vowel of English alphabet}

3.3.3 Descriptive FormIn this method, the elements in the set are verbally described within the curly brackets

Example:N= {natural numbers from 1 to 100}V= {vowels of English alphabet}

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3.4 Types of SetsFollowing are the types of sets

3.4.1 Finite SetAsetwithafiniteorcountablenumberofelementsiscalledafiniteset.Example:V= {a, e, i, o, u}

3.4.2 Empty or Null SetA set which contains no elements is called an empty set or a null set. An empty set is denoted as { } or .Example:A= { } A=

3.4.3 SubsetSet P is a subset of a set Q, symbolised by P Q, if and only if all the elements of set P are also the element of set Q.

Note:every set is a subset of itselfnull set is a subset of every set

Example:Let A = {1, 2, 3}Then { }, {1}, {2}, {3},{1,2},{1,3},{2,3}.{1,2,3} are the subsets of the set A={1,2,3}.

3.4.3.1 Proper SubsetSet G is a proper subset of H, symbolised by G H, if and only if all the elements of set G is elements of set H and set G set H.

That is set H must contain at least one element not in set G.ExampleConsider above example, {},{1},{2},{3},{1,2},{1,3},{2,3} are the proper subset of A={1,2,3}.

3.4.3.2 Improper SubsetSet S is an improper subset of T, symbolised by S T , if and only if all the elements of set S are the elements of set T and set S = set T.ExampleIn the above example {1,2,3} is an improper subset of set A={1,2,3}

3.4.4 Infinite Set

Neitheranemptysetnorafinitesetiscalledaninfiniteset.Suchsetwillcontaininfinitelymanyelements.ExampleN= {1, 2, 3...}Z= {...-3,-2,-1, 0,1,2,3...}


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3.4.5 Disjoint SetsTwo sets are said to be disjoint sets if they do not have any common elements.Example IfA={x: x is an even number}B={y: y is an odd number}Then A and B will be disjoint sets.

3.4.6 Overlapping SetsTwo sets are said to be overlapping sets, if they have some common elements.ExampleA= {1, 2, 3, 4, 5}B= {3, 4, 5, 6, 7}

These sets are overlapping sets as the elements 3, 4 and 5 are common to both set.The element common to both sets is called the intersection of the two sets and is denoted as A B.On the other hand, the addition of two sets is called the union of two sets and is denoted by A B.

3.4.7 Universal SetA universal set is the set that contains the element of all the sets under consideration. It is usually denoted by U, S or .ExampleA={4,7,8,9}, B={-4,-2,0,1,4,7,10}.The set of Integers, I= {...,-2,-1, 0, 1, 2...} will be the universal set for A and B.

3.4.8 Equal SetTwo sets are said to e equal if they contain the same elements.

ExampleA= {2, 4, 6, 8} B={x: x is an even number between 1 and 9} are equal set as they contain the same elementsP= {1, 3, 6} Q= {6, 1, 3} is also an equal set.

3.4.9 Complement SetGiven a set A, the complement set is the set that contains elements not belonging to A and is denoted by A. The union (will be discussed in operations on set) of the given set and its complement will give the universal set. That is A A = U.

ExampleA={x: x is a Mathematics book in the Library}A ={y: y is not a Mathematics book in the Library}And the universal set in this case is U= the set of all books in the Library}

3.4.10 Equivalent SetTwo sets are said to be equivalent set if number of elements in one set is equal to the number of elements in the other set. The number of elements in a set is known as Cardinal number of the set.

So, if the cardinal numbers of two sets are equal then those two sets are said to be equivalent set. Cardinal number of the set A is represented by n (A).ExampleA= {1, 2, 3, 4} B= {a, b, c, d}

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Here, n (A) =4 and n (B) =4So the sets A and B are said to be Equivalent sets.

3.5 Illustration of Various SetsLet us consider three sets A, B and C in the rule form as A= {x: x are the vowels}, B= {x: x are the consonants} C={x:xisthefirst10alphabetsofEnglish}

In tabular formA={a,e,i,o,u}, B={b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z},C={a,b,c,d,e,f,g,h,i,j}

set A and B are disjoint sets as they no elements of A is present in B and vice versa.set A and C are overlapping sets as both contains the element a, e, i in common.set B and C are also overlapping set as both contain the element b, c, d, f, g, h, j in common.set of all alphabets of English is the universal set for A, B, C. U= {a, b, c, d... z}set A is a complement of set B as it contains elements not contained in set A and union of set A and set B will give the Universal set U.

3.6 Basic Operations on SetsFollowing are the basic operations on sets

3.6.1 Intersection of Two SetsThe intersection of two sets is the set of elements common to both the given sets. The intersection of two sets A and B is denoted as A B.

Innotationform,wecandefinetheintersectionoftwosetsAandBasA B = {x: x A, x B}.

If, A B = , then A and B are said to be disjoint sets. If A B ,then A and B are called overlapping sets.

ExampleGivenA= {1, 2, 3, 4, 5}B= {4, 5, 6, 7, 8}Then A B = {4, 5}

3.6.2 Union of Two SetsThe union of the two sets is the set containing the elements belonging to A and also the elements belonging to B.The union of these sets is denoted as A B.Innotationform,wecandefinetheunionofthetwosetsasA B = {x: x A, x B, x A B}.

ExampleA= {

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