Business Mathematics II BCom

1
Business MathematicsII BCom
2
Units

• Elements of Set theory
• Indices and Logarithms
• Commercial Arithmetic
• Differential Calculus
• Determinants

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3
Elements of Set Theory

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4
Introduction

• Understanding set theory helps people to
• see things in terms of systems
• organize things into groups
• begin to understand logic

5
Definition

• Set A well-defined collection of distinct
  objects is called a set.
•        
• Element An element is an object contained in a
  set.
• Notation of Sets Capital letters are usually
  used to denote or represent a set.

6
Definition

• Representation of Sets There are two methods of
  representing a set. (i) Roster Method (ii) Set
  builder form.
• Finite and Infinite Sets A set is finite if it
  contains a specific number of elements.
  Otherwise, a set is an infinite set.

7
Definition

• Null Set or Empty Set or Void Set A set with no
  elements is an empty set.
•        
• Singleton Set or Singlets A set consisting of a
  single element is called a singleton set or
  singlet. The cardinality of the singleton set is
  1.
• Equivalent Sets Two finite sets A and B are said
  to be equivalent sets if cardinality of both sets
  are equal i.e. n (A) n (B).

8
Definition

• Equal Sets Two sets A and B are said to be equal
  if and only if they contain the same elements
  i.e. if every element of A is in B and every
  element of B is in A. We denote the equality by A
  B.
• Cardinality of a Set A The number of elements in
  a finite set A, is the cardinality of A and is
  denoted by n(A).

9
Definition

• Universal Set In any application of the theory
  of sets, the members of all sets under
  consideration usually belong to some fixed large
  set called the universal set.
•       
• Power Set The family of all subsets of any set S
  is called the power set of S. We denote the power
  set of S by P (S).

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Assignment

• Write all definitions.

11
Symbols

• Upper case designates set name
• Lower case designates set elements
• enclose elements in set
• ? or is (or is not) an element of
• ? is a subset of (includes equal sets)
• ? is a proper subset of
• ? is not a subset of
• ? is a superset of
• or such that (if a condition is true)
• the cardinality of a set

12
Symbols

•  Sa, b, c refers to the set whose elements are
  a, b and c.
• a?S means a is an element of set S.
• d?S means d is not an element of set S.

13
Symbols

•  x ?S P(x) is the set of all those x from S
  such that P(x) is true.
• E.g., Tx ?Z 0ltxlt10 .
• Notes
• a,b,c, b,a,c, c,b,a,b,b,c all represent the
  same set.
• Sets can themselves be elements of other sets,
  e.g., S cat, rat, Pot, Pan,

14
Roster method and rule method

• Roster method The method of specifying a set
  consists of surrounding the collection of
  elements with braces.

15
Roster method and rule method

• Set Builder method This has the general form
• variable descriptive statement .
• The vertical bar (in set builder notation) is
  always read as such that.
• Set builder notation is frequently used when the
  roster method is either inappropriate or
  inadequate

16
Types of sets

• Subset If every element of Set A is also
  contained in Set B, then Set A is a subset of Set
  B
• A is a proper subset of B if B has more elements
  than A.

17
Union and Intersection

• Intersection The intersection of two sets A and
  B is the set containing those elements which are
  elements of A and elements of B.
• Written as A ? B

18
Union and Intersection

• Union The union of two sets A and B is the set
  containing those elements which are elements of A
  or elements of B.
• Written as A ? B

19
Algebraic Properties

• Commutative operations
• A ? B B ? A
• A n B B n A
• Distributive Law
• A n ( B ? C ) (A n B) ? (A n C)
• A ? ( B n C ) (A ? B) n (A ? C)

20
Algebraic Properties

• Associative Law
• A n ( B n C ) (A n B) n C
• A ? ( B ? C ) (A ? B) ? C
• Other Properties
• A ? ? A A n ? ?
• A ? A A A n A A

21
Assignment

• Write proofs for
• A n B B n A
• A ? ( B n C ) (A ? B) n (A ? C)
• A ? ( B ? C ) (A ? B) ? C

22
Subsets

• Subset If every element of Set A is also
  contained in Set B, then Set A is a subset of Set
  B
• A is a proper subset of B if B has more elements
  than A.

23
Complements

• Let U be a universal set. The complement of a set
  A is defined to be the set of all elements which
  are in U and not in A. The complement of A is
  denoted by A or A or Ac.
• (i.e.) A x x?U, x?A

24
Difference of two sets

• The difference of sets A and B is defined to be
  the set which contains all those elements in A
  which are not in B. The difference of set A and B
  is denoted by A B
• (i.e.) A-B x x?A, x?B
• Similarly B-A x x?B, x?A
• Note
• A-B ? B-A

25
Family of sets

• Suppose A and B are sets. Then A is called a
  subset of B A ? B
• iff every element of A is also an element of B.
• Symbolically,
• A ?B ? ?x, if x?A then x ?B.
• A ? B ? ?x such that x?A and x?B.

26
Venn diagram

• Venn Diagram
• A Venn diagram is a pictorial representation of
  sets by set of points in the plane. The universal
  set U is represented pictorially by interior of a
  rectangle and the other sets are represented by
  closed figures viz circles or ellipses or small
  rectangles or some curved figures lying within
  the rectangle.

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Venn diagram

• Venn diagrams show relationships between sets and
  their elements

U
A
B
28
Venn diagram

• A is a subset of B and is represented as shown in
  the venn diagram.

29
De-Morgans law

• (A?B) AnB
• (AnB) A?B

30
Assignment

• Prove (AnB) A?B using Venn diagram.

31
Further reading

• Business Mathematics
• Dr. P.R. Vittal
• Business Mathematics
• Dr. B.M. Aggarwal
• http//www.tutorvista.com/content/math/number-theo
  ry/sets/setsindex.php

32
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33
Indices and Logarithms

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34
Positive and Fractional

• Laws of Indices  
• If m and n are positive integers, and a?0 then  
• Product Law am.an am n  
• Quotient Law am/an am - n
• Power Law (am)n amn

35
Operation with power function

• (ab)m am . bm  
• am/n(am) 1/nnvam
• ao 1
• a-11/a

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Assignment

• If x yz, y zx, z xy show that xyz1
• Show that
• is independent of a and n.

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Logarithms - Definition

• Definition The logarithm of a positive number to
  a given base is the index of the power to which
  the base is raised to equal the number.
• Consider N ax
• N is a positive number, is the base and is the
  index of the power. From the definition we can
  write the equation N ax as

38
Laws of logarithms

• Product Rule
• Quotient Rule

39
Laws of logarithms

• Power Rule

40
Change of base - formula

• Rule of Change of base

41
Assignment

• Find the value of
• Prove that

42
Common logarithms Natural logarithms

• Common Logarithms
• Logarithms with base 10 are called common
  logarithms
• Natural Logarithms
• Logarithms with base e are called Napier or
  Natural logarithms. The value of e lies between 2
  and 3.

43
Characteristic Mantissa

• Characteristics and Mantissa
• The logarithm of a number contains two parts
• 1) the integral part and 2) the decimal part. The
  integral part is called characteristics and the
  decimal part is called the mantissa

44
Rule to write

• For a number greater than one the characteristic
  is the number, one less than the number of digits
  before the decimal point and is positive.

45
Rule to write

• For a number less than one the characteristic is
  the number one more than the number of zeros
  between the decimal point and the first non-zero
  digit in the number expressed as a decimal and it
  is negative.

46
Assignment

• Find the number of digits in 715
• When 2-53-4 is expressed as a decimal find the
  position of the first significant figure.

47
Further reading

• Business Mathematics
• Dr. P.R. Vittal
• Business Mathematics
• Dr. B.M. Aggarwal

48
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49
Commercial Arithmetic

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50
SIMPLE INTEREST AND COMPOUND INTEREST
51
Interest

• Interest is the cost of borrowing money. 
• An interest rate is the cost stated as a percent
  of the amount borrowed per period of time,
  usually one year.

52
Simple Interest

• Simple interest is calculated  on  the original
  principal only.

53
Compound Interest

• Compound interest is calculated each period on
  the original principal  and all  interest
  accumulated during past periods. 
• Although the interest may be stated as a yearly
  rate, the compounding periods can be yearly, semi
  annually, quarterly, or even continuously.

54
Compound Interest
55
Compound Interest
56
Compound Interest
57
Normal rate Effective rate

• When the interest is compounded half-yearly or
  quarterly the interest which becomes due at the
  end of the year on a loan is called the Effective
  Rate of Interest.

58
Normal rate Effective rate

• The given rate for calculating interest for half-
  yearly or quarterly is called the Nominal Rate.

59
Normal rate Effective rate
60
Depreciation
61
Present value

• When Compound interest is considered.
• When Simple interest is considered.

62
Assignment

• The simple interest on a certain principal for 5
  years is Rs.360 and the interest is 9/25 of the
  principal. Find the principal and the interest
  rate.
• The population of a country increases every year
  by 2.4 percent of the population at the beginning
  of the year. In what time will the population
  double itself? Answer to the nearest year.
• Compute the interest on Rs.1000 for 10 years at
  4 per annum, the interest being paid annually.

63
Discounting of bills

• To honour any transaction it is necessary to have
  documents like bill of exchange or promissory
  note according to which the debtor has to pay to
  the creditor a specified sum of money on a
  specified date.

64
Discounting of bills

• Two kinds of bills
• Bill of exchange after date
• Bill of exchange after sight

65
Face value of bills

• Face value is the value of a coin, stamp or paper
  money, as printed on the coin, stamp or bill
  itself by the minting authority. While the face
  value usually refers to the true value of the
  coin, stamp or bill.

66
Bankers discount

• True Discount The interest on the present value
  is called True Discount
• T.D A P or T.D Pni
• Where A Total Amount
• P Principal
• Bankers Discount The interest on sum due is
  called Bankers Discount.
• B.D Ani
• Where A Amount
• n number of years
• i- interest percent

67
Bankers discount

• Bankers Gain B.D T.D
• Or
• Bankers gain is interest on true discount

68
Normal due date legal due date

• The date on which the bill of exchange becomes
  due is called the nominal due date or simply due
  date.
• There is a legal custom to allow 3 days of grace
  from the due date to encash a bill. If these days
  are added to the due date, we get the date on
  which the bill becomes legally due.

69
Calculation of period for bankers discount and
true discount

• Find the date of drawing the bill.
• Find the date of maturity.
• Find the date on which the bill was discounted.
• Count the number of days from the date of
  discount to the date of maturity.
• Add three days grace period to this to give the
  legal due date.

70
Assignment

• A bill of Rs.3,225 was drawn on 3rd February 1965
  at 6 months due date and discounted on 13th March
  1965 at the rate of 8 per annum. For what sum
  was the bill discounted and how much did the
  banker gain in this?
• The difference between true and bankers discounts
  on a certain bill due in 4 months is 50 paise. If
  the rate of interest is 6 percent, find the
  amount of the bill.

71
Further reading

• Business Mathematics
• Dr. P.R. Vittal
• Business Mathematics
• Dr. B.M. Aggarwal

72
End of Unit

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73
Differential Calculus (excluding trigonometric
functions)

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74
Introduction

• Differential calculus can be considered as
  mathematics of motion, growth and change where
  there is a motion, growth, change. Whenever there
  is variable forces producing acceleration,
  differential calculus is the right mathematics to
  apply. Application of derivatives are used to
  represent and interpret the rate at which
  quantities change with respect to another
  variable. Differential equations are powerful
  tools for modeling data.

75
Introduction

• A differential equation is an equation involving
  derivatives of an unknown function and possibly
  the function itself as well as the independent
  variable.
• The order of a differential equation is the
  highest order of the derivatives of the unknown
  function appearing in the equation

76
Rules

• If y is a function of x, that is y f(x), we
  write its gradient function as .
• if y xn, then
• If yc, then
• Where c is a constant

77
Rules

78
Sum rule

79
Product rule

• It is appropriate to use this rule when you want
  to differentiate two functions which are
  multiplied
• together.
• If u and v are functions of x, then

80
Quotient rule

• If u and v are functions of x, then
• In the rule which follows we let u stand for the
  function in the numerator and v stand for the
• function in the denominator.

81
Function of function rule (simple problems only)

• Let y be a function of u and u be a function of
  x. Then
• Let y be a function of u, u is a function of v
  and v is a function of x. Then

82
Assignment

• Differentiate the following with respect to x.

83
Maxima and minima (Single variable cases)

• The diagram below shows part of a function y
  f(x).

84
Maxima and minima (Single variable cases)

• The point A is a local maximum and the point B is
  a local minimum. At each of these points the
  tangent to the curve is parallel to the x-axis so
  the derivative of the function is zero. Both of
  these points are therefore stationary points of
  the function. The term local is used since these
  points are the maximum and minimum in this
  particular region. There may be others outside
  this region.

85
Maxima and minima (Single variable cases)

• The rate of change of a function is measured by
  its derivative. When the derivative is positive,
  the function is increasing, when the derivative
  is negative, the function is decreasing. Thus the
  rate of change of the gradient is measured by its
  derivative, which is the second derivative of the
  original function. In mathematical notation this
  is as follows.

86
Maxima and minima (Single variable cases)

• At the point (a, b)
• If and then
  point(a,b)
• is a local maximum.
• If and then point(a,b)
  is a
• Local minimum.

87
Methods of integral calculus rules(excluding
integration by parts or fractions)

• There are two general rules
• Few other rules

88
Methods of integral calculus rules(excluding
integration by parts or fractions)

• Other rules

89
Methods of integral calculus rules(excluding
integration by parts or fractions)

• Other standard results

90
Assignment

• Find the minimum average cost for the average
  cost function and show
  that at the minimum average cost, marginal cost
  and average cost are equal.
• Integrate the following with respect to x.
• 3x37x2-2x1
• (1-x)3

91
Further reading

• Business Mathematics
• Dr. P.R. Vittal
• Business Mathematics
• Dr. B.M. Aggarwal
• http//www.tutorvista.com/content/math/calculus/di
  fferentiation/differentiationindex.php

92
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93
DETERMINANTS
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94
Introduction

• Definition of a Matrix A rectangular array of
  entries is called a Matrix. The entries may be
  real, complex or functions.   The entries are
  also called as the elements of the matrix.   The
  rectangular array of entries are enclosed in an
  ordinary bracket or in square bracket. Matrices
  are denoted by capital letters.

95
Determinant

• Let A aij be a square matrix.
• A determinant formed by the same array of
  elements of the square matrix A is called the
  determinant of the square matrix A and is denoted
  by the symbol det.A or A.
• The determinant of a square matrix will be a
  scalar quantity. i.e., with a determinant we
  associate a definite value, whereas a matrix is
  essentially an arrangement of numbers and has no
  value.

96
Determinant Properties

• If the rows and columns of a determinant are
  inter-changed, the value remains unaltered.
•  
• If any two rows (columns) of a determinant are
  identical, its value of the determinant is zero.
•  
• If any two rows (columns) of a determinant are
  interchanged, the value of the determinant is
  (-1) times the original determinant.

97
Determinant Properties

• If all the elements of one row (column) of a
  determinant is multiplied by k, the value of the
  new determinant is k times the original
  determinant.
•  
• If to any row or column of a determinant, a
  multiple of another row or column is added, the
  value of the determinant remains the same.
•  
• If some or all the elements of a row (or column)
  of a determinant are expressed as sum of two (or
  more) terms, then the determinant can be
  expressed as sum of two or more determinants.

98
Determinant Properties

• The sum of the products of the elements of any
  row (column) with their corresponding cofactors
  is equal to the value of the determinant.
•  
• The sum of the products of the elements of any
  row (column) and the cofactors of the
  corresponding elements of any other row (column)
  is zero.

99
Matrices - types

• Row Matrix
• Column Matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity or Unit Matrix

100
Matrices - types

• Null Matrix or Zero matrix
• Upper triangular matrix
• Lower triangular matrix
• Transpose of a matrix
• Symmetric matrix
• Skew-symmetric matrix

101
Assignment

• Write all different types of Matrix

102
Matrices Addition

• Only matrices of the same order can be added or
  subtracted. The result from the sum or difference
  is then determined by adding or subtracting the
  corresponding elements.

103
Matrices Addition
104
Matrix - Multiplication

• Two matrices A and B are conformable to matrix
  multiplication only if the number of columns in A
  equals the number of rows in B.
• AB has the same number of rows as A and the same
  number of columns as B.
• The ijth element of AB is the inner product of
  the ith row of A and jth column of B.

105
Matrix - Multiplication

106
Assignment

• If A
• B
• Find 1) AB
• 2) A-B

107
Assignment

• If A
• B
• Find 1) AB
• 2) BA

108
Matrix inversion

• The inverse of a square matrix A is a matrix
  whose product with A is the identity matrix I.
• The inverse matrix is denoted by A-1.

109
Matrix inversion

• The concept of dividing by A in matrix algebra
  is replaced by the concept of multiplying by the
  inverse matrix A-1.
• An inverse matrix A-1 has the following
  properties
• A-1 A A A-1 I
• A-1 unique for given A.

110
Matrix inversion

• The (1) and (1) factors in the expansion are
  decided on according to the following rule
• If A is written in the form A (aij), the
  product of aij and its minor in the expansion of
  determinant A is multiplied by (1)ij.

111
Matrix inversion

• Therefore, because the element 1 in the example
  is the element a11, its product with its minor is
  multiplied by (1)11 1. For element 2, which
  is a12, its product with its minor is multiplied
  by (1)12 1

112
Matrix inversion

• The determinant of a square matrix of
• order n, (that is, Ann (aij) i, j 1, 2, ,
  n) is the sum of all possible products of n
  elements of A such that each product has one and
  only one element from every row and column of A,
  the sign of a product being (1)ij.

113
Matrix inversion

• The determinant of a square matrix A,
• denoted by A, is a polynomial of the
• elements of a square matrix.
• It is a scalar.
• It is the sum of certain products of the elements
  of the matrix from which it is derived, each
  product being multiplied by 1 or 1 according to
  certain rules

114
Solving a system of linear equation using matrix
inversion

• Consider the linear equation
• a1xb1yc1z d1
• a2xb2yc2z d2
• a3xb3yc3z d3
• Denote A

115
Solving a system of linear equation using matrix
inversion

• X
• E
• Then AX E
• X A-1E

116
Rank of a Matrix

• The rank of a matrix A is the order of its
  largest non-zero minor.
• Minor The determinant of a submatrix of order r
  of a given matrix will be called a minor of order
  r of the matrix.

117
Elementary Row Operations

• Operations that can be performed without altering
  the solution of a linear system.
• Change any two rows.
• Multiply every element in a row by a non-zero
  constant.
• Add elements of one row to corresponding elements
  of another row.

118
Testing consistency of equations

• Consider the linear equation
• a1xb1yc1z d1
• a2xb2yc2z d2
• a3xb3yc3z d3
• Denote A and B

119
Testing consistency of equations

• The augmented matrix
• AB
• Find the rank of matrix A

120
Testing consistency of equations

• Find the rank of augmented matrix AB using row
  transformation.
• Using row transformation, transform the augmented
  matrix AB to an Echelon matrix.
• Write the system of equations corresponding to
  the resulting matrix.
• Use back-substitution to find the systems
  solution

121
Assignment

• Test the consistency for the following
• 2x-yz4
• xyz3
• 3x-y-z1

122
Further reading

• Business Mathematics
• Dr. P.R. Vittal
• Business Mathematics
• Dr. B.M. Aggarwal
• http//www.tutorvista.com/content/math/number-theo
  ry/matrices-and-determinants/matrices-and-determin
  antsindex.php

123
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124
End of Business Mathematics

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