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UNIT - I

Lesson 1 - Set theory and Set Operations
Contents:
1.1 Aims and Objectives 1.2 Sets and elements 1.3 Further set concepts 1.4 Venn Diagrams 1.5 Operations on Sets 1.6 Set Intersection 1.7 Let – us Sum Up 1.8 Lesson – End Activities 1.9 References

1.1 Aims and Objectives
This Lesson introduces some basic concepts in Set Theory, describing sets, elements, Venn diagrams and the union and intersection of sets.

1.2 Sets and elements
Sets of objects, numbers, departments, job descriptions, etc. are things that we all deal with every day of our lives. Mathematical Set Theory just puts a structure around this concept so that sets can be used or manipulated in a logical way. The type of notation used is a reasonable and simple one. For example, suppose a company manufactured 5 different products a, b, c, d, and e. Mathematically, we might identify the whole set of products as P, say, and write: P = (a,b,c,d,e) which is translated as 'the set of company products, P, consists of the members (or elements) a, b, c, d and e. The elements of a set are usually put within braces (curly brackets) and the elements separated by commas, as shown for set P above. A mathematical set is a collection of distinct objects, normally referred to as elements or members. Sets are usually denoted by a capital letter and the elements by small letters. Example 1 (Illustrations of sets)

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2 a) The employees of a company working in the purchase department could be written as: P = (Jones, Wilson, Gopan, Smith, Hari) b) The warehouse locations of a large supermarket chain could be written as: W = (Mumbai, Delhi, Bangalore, Chennai, Triandrum, Kochi)

1.3 Further set concepts
a) Subsets. A subset of some set A, say, is a set which contains some of the elements of A. For example, if A = (h,i,j,k,l), then: X = (i,j,l) is a subset of A Y = (h,1) is a subset of A Z = (i,j) is a subset of A and also a subset of X. b) The number of a set. The number of a set A, written as n[A], is defined as the number of elements that A contains. For example, if A = (a,b,c,d,e), then n[A] = 5 (since there are 5 elements in A); if D = (Sales, Purchasing, Inventory, Payroll), then n[D] = 4. c) Set equality. Two sets are equal only if they have identical elements. Thus, if A = (x, y, z) and B = (x, y, z), then A = B. d) The Universal Set. In some problems in involving sets, it is necessary to consider one or more sets under consideration as belonging to some larger set that contains them. For example, if we were considering the set of skilled workers (S, say) on a production line, it might be convenient to consider the universal set (U, say) as all of the workers on the line. In other words, where a universal set has been defined, all the sets under consideration must necessarily be subsets of it. e) The complement of a set. If A i s any set, with some universal set U defined, the complement of A, normally written as A', is defined as 'all those elements that are not contained in A but are contained in U'. For the example of the workers on the production line (given in d above), S was specified as the set of skilled workers within the universal set of all workers on the line. Therefore, S' would be all the workers that were not skilled. i.e. the set of unskilled workers.

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1.4 Venn Diagrams
. A Venn diagram is a simple pictorial representation of a set. For example, if M = (a,b,c,d,e,f,g) then we could represent this information in the form of a Venn diagram as in Figure 1.1 Figure 1.1 b a f c g e 6 Venn diagrams are useful for demonstrating general relationships between sets. For example, if a firm maintains a fleet of 7 cars, we might write A = (1,2,3,4,5,6,7) (each car being numbered for convenience). If also it was important to identify those cars of the fleet that were being used by the directors, we might have D = {3,5 ). i.e. Cars 3 and 5 are director's cars. This situation could be represented in Venn diagram form as in Figure 1.2. This diagram nicely demonstrates the fact that D is a subset of A, which normally means that n[D] < n[A]. In this case n[D] = 2 and n [A] = 7. d 2 5 M 1 D 3 7 4 Figure 1.2 A

1.5 Operations on Sets
In ordinary arithmetic and algebra there are four common operations that can be performed; namely, addition, subtraction, multiplication and division. With sets, however, just two operations are defined. These are set union and set intersection. Both of these operations are described, with examples, in the following sections.

The Union of two sets A and B is written as AÈB and defined as that set which contains all the elements lying within either A or B or both. For example, if A = (c,d,f,h,j) and B = (d,m,c,f,n,p), then the union of A and B is AÈB = (c,d,f,h,j,m,n,p), these being the elements that lie in either A or B. So that any element of A must be an element of AÈB; similarly any element of B must also be an element of AÈB. Set union for three or more sets is defined in an obvious way. That is, if A, B and C are any three sets, AÈBÈC is the set containing all the elements lying within (i) anyone of A, B or C, (ii) any two of them or (iii) all three. Example 2 (To demonstrate set union)

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4 If A = (m,n,o,p); B = (m,o,p,q); C = (m,p,r); and the universal set is defined as U = (k,l,m,n,o,p,q,r,s), then: a) AÈB = (m,n,o,p,q) b) AÈC = (m,n,o,p,r) c) BÈC = (m,o,p,q,r) d) AÈBÈC = (m,n,o,p,q,r) e) (AÈB)' = (k,l,r,s), which is describing all the elements that are not in AÈB but are in the universal set U.

1.6 Set Intersection
The intersection of two sets A and B is written as AÇ B and defined as that set which contains all the elements lying within both A or B. For example, if A = (a,b,c,d,f,g,) and B = (c,f,g,h,j), then the intersection of A and B is AÇ B = (c,f,g), since these are the elements that lie in both sets. The intersection of three or more sets is a natural extension of the above. If P, Q and R are any three sets then PÇ QÇ R is the set containing all the elements that lie in all three sets. Any combinations of union and intersection can be used with sets. For, example, if X and Y are the sets specified above and Z = (d,f,g,j). then: (XÇy) ÈZ = (c,f,g) È(d,f,g,j) =(c,d,f,g,j) which can be described in words as 'the set of elements that are in either both of X and Y or in Z’.

Example 3 (To demonstrate set intersection) If A=(m,n,o,p};B=(m,o,p.q);C=(n,q,r);with a universal set defined as (k,l,m,n,o,p,q,r,s). Then: a) AÇB = (m,o,p), since a1l these elements are in both sets. Similarly, b) AÇ e = (n) c) BÇC = (q). d) AÇ BÇ C has no elements, is sometimes called the empty set and can be written AÇ BÇC = {}. Note n[{}]=0. e) (AÇ B)' = (k,l,n,q,r,s) is the complement of AÇB and is the set of all elements that are NOT in both A and B. f) (AÈB)ÇC =(m,n,o,p,q)Ç(n,q,r)= (n.q) is the set of all elements that are in A o r B AND ALSO in C.

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5 Example 4 (The union and intersection of given sets) Question In a particular insurance life office, employees Smith, Jones, Williams and Brown have 'A’ levels, with Smith and Brown also having a degree. Smith, Melville, Williams, Tyler, Moore and Knight are associate members of the Chartered Insurance Institute (ACII) with Tyler, and Moore having 'A’ levels. Identifying set A as those employees with 'A' levels, set C as those employees who are ACII and set D as graduates: a) Specify the elements of sets A, C and D. b) Draw a Venn diagram representing sets A, C and D, together with their known elements. c) What special relationship exists between sets A and D? d) Specify the elements of the following sets and for each set, state in words what information is being conveyed. i. AÇC ii. DÈC iii. DÇC e) What would be a suitable universal set for this situation? Answer a) A = (Smith, Jones, Williams, Brown, Tyler, Moore); C = (Smith, Melville, Williams, Tyler, Moore, Knight); D = (Smith, Brown) b) The Venn diagram is shown in Figure 1.3.

A C Jones Moore D Brown Smith Williams Tyler Melville Knight

Figure 1.3

c) From the diagram, it can be seen that D is a subset of A. d) This information can be obtained either from the Venn diagram or from the sets listed in, a) above. i. AÇC = (Williams, Tyler, Smith). This set gives the employees who have both ‘A' levels and are ACII. ii. DÈC = (Brown, Smith, Williams, Tyler, Melville, Knight). This set gives the employees who are either graduates or ACII.

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6 iii. DÇC = (Smith). This set gives the single employee who is both a graduate and ACII qualified. e) A suitable universal set for this situation would be the set of all the employees working in the Life office.

1.7 Let us Sum Up
This Lesson presented described about the set, set theory, Venn Diagrams, and its applications. A set is a collection of distinct objects, called elements, which are normally enclosed within brackets and separated by commas. Venn diagram is a pictorial representation of one or more sets. The Union and Intersection of sets were also discussed in detail. Some examples to understand the concept is also given in the Lesson.

1.8 Lesson – End Activities
1. Define set, subset. 2. Give the purpose of drawing Venn diagrams.

1.9 References
Navaneethan. P. – Business Mathematics.

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Lesson 2 - Functions and Co-ordinate Systems
Contents: 2.1 Aims and Objectives 2.2 Definitions 2.3 Types of Functions 2.4 Solution of Functions 2.5 Business Applications 2.6 Let us Sum Up 2.7 Lesson – End Activities 2.8 Reference

2.1 Aims and OBjectives
For decision problems which use mathematical tools, the first requirement is to identify or formally define all significant interactions or relationships among primary factors (also called variables) relevant to the problem. These relationships usually are stated in the form of an equation (or set of equations) or inequations. Such type of simplified mathematical relations help the decision- maker in understanding (any) complex management problems. For example, the decision- maker knows that demand of an item is not only related to price of that item but also to the price of the substitutes. Thus if he can define specific mathematical relationship (also called model) that exists, then the demand of the item in the near future can be forecasted. The main objective of this unit is to study mathematical relationships (or functions) in the context of managerial problems.

2.2 Definitions
Variable A variable is something whose magnitude can vary or which can assume various values. The variables used in applied mathematics include: sale, price, profit, cost, etc. since magnitude of variables can vary, therefore these are represented by symbols (such as x,y,z etc) instead of a specific number. In applied mathematics a variable is represented by the first letter of its name, for example p for price or profit; q for quantity, c for cost; s for saving or sales; d for demand and so forth. When we write X = 5, the variable takes specific value. Variables can be classified in a number of ways. For example, a variable can be discrete (suspect to counting, e.g. 2 houses, 3 machines etc.), or continuous (suspect to measurement, e.g. temperature, height etc.).

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8 Constant and Parameter A quantity that remains fixed in the context of a given problem or situation is called a constant. An absolute (or numerical) constant such as 2, π, e, etc. retains the same value in all problems whereas an arbitrary (or parametric) constant or parameter retains the same value throughout any particular problem but may assume different values in different problems, such as wage rates of different category of labourers in an industrial unit. The Absolute or numerical value of a constant ‘b’ is denoted by |b| and means the magnitude of ‘b’ regardless of its algebraic sign. Thus |b| = |-b | or |+b|.

Functions We come across situations in which two or more variables are related to each other. For example, demand (D) of a commodity is related to its price (p). It can be mathematically expressed as D = f(p) (2-1) This relationship is read as “demand is function of price” or simply “f of p”. it does not mean D equals f times p. This mathematical relationship has two variables, D and p. these are called variables because they can take on different numerical values. Let us now consider a mathematical relationship that contains three variables. Assume that the demand (D) of a commodity is related to the price (p) per unit of the commodity, and the level of advertising expenditure (A). then the general relationship among these variables can be expressed as D = f(p,A) (2-2) The functional notations of the type (2-1) and (2-2) are meant to give a general idea that certain variables are, somehow, related. However for making managerial decisions, we need a specific and explicit, not a general and implicit relationship among selected variables. For example, for the purpose of finding the value of demand (D), we make the general relationship (2-2) more specific as shown in (2-3). D = 4+3p-2pA+2A2 (2-3)

Now for any given values of p and A, the value of D can be calculated using the relationship (2-3). This means that the value of D depends on the values of p and A. Hence D is called the dependent variable and p and A are called independent variables. In this case, it may be noted that we have established a rule of correspondence between the dependent variable and independent variable (s). That is as soon as values are assigned to the independent variables (s), the corresponding unique value for the dependent variable is determined by the given specific relationship. That is why a function is sometimes defined as a rule of correspondence between variables. The set of values given to independent variable is called

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9 the domain of the function while the corresponding set of values of the dependent variable is called the range of the function. Other examples of functional relationships are as follows: i) ii) iii) iv) v) vi) the distance (d) covered is a function of time (T) and speed (s), i.e. d = f (T,s). Sales volume (v) of the commodity is a function of price (p), i.e. V = f(p). Total inventory cost (T) is a function of order quantity (Q), i.e. T = f(Q). The volume of the sphere (v) is a function of its radius ®, i.e. V = f® or V = 4/3 π r3 The extension (y) of a spring is proportional to the weight (m) (Hooke’s law), i.e. Y m or Y = km. The net present value (y) of an investment is a function of net cash flows (Ct ) in different time periods, project’s initial cash outlay (B), firm’s cost of capital (P) and the life of the project (N), i.e. y = f(Ct , B,P,N).

The following example will illustrate the meaning of these terms. Example 1 Suppose an industrial worker gets Rs. 50 per day. If he works for 25 days in a particular month, then his total wage for this month is 50 x 25 = Rs. 1250. During some other month he may have worked a total of only 24 days, then he would have earned Rs. 1200. Thus, the total wages of the worker, assuming no overtime, can always be calculated as follows: Total wages = 25 X number of days worked Let, T = total wages D = number of days worked Then, T = 50 D. This represents the relationship between total wages and number of days worked. In general, the above relationship can also be written as: T = KD Where K is a constant for particular class of worker (s), to be assigned or determined in a specific situation. Since the value of K can vary for a specific situation, problem or context therefore it is called a parameter, whereas constants such as pi (denoted by π) which has approximate value of 3.1416 remains same from one problem context to another are called absolute constants. Quantities such as T and D which can assume various values in a given problem are called variables. Exercise 1. Find the domain and range of each of the following functions a) Y = 1/x-1 b) Y = x; y 0 c) Y = 4-x; y 0 2. Let 4p+6q = 60 be an equation containing variables p (price) and q (quantity). Identify the meaningful domain and range for the given function when price is considered as independent variable.

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2.3 Types of Functions
In this Section some different types of functions are introduced. Linear Functions: A linear function is one in which the power of independent variable is 1, the general expression of linear function having only one independent variable is: Y = f(x) = a + bx Where a and b are given real numbers and x is an independent variable taking all numerical values in an interval. A function with only one independent variable is also called single variable function. Further, a single- variable function can be linear and non-linear. For example, Y = 3+2x, (linear single-variable function) And Y = 2+3x-5x2 +x2 , ( non- linear single-variable function) A liner function with one variable can always be graphed in a two dimensional plane (or space). This graph can always be plotted by giving different values to x and calculating corresponding values of y. the graph of such functions is always a straight line. Example 2 Plot the graph of the function, y = 3+2x For plotting the graph of the given function, assigning various values to x and then calculating the corresponding values of y as shown in the table below:

X Y

0 3

1 5

2 7

3 9

4 11

5 13

… …

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11 The graph of the given function is shown in Figure 1.4 y 13 11 (4,11) y =3+2x 9(3,9)

7-

(2,7)

5-

(1,5)

30

(0,3) |

| 1

| 2

| 3

| 4

| 5

x

Figure 1.4 A function with more than one independent variable is defined, in general, form, as: Y=f(x1 ,x2 ,….,xn ) = a0 +a1 x+a2 x2 +…+an xn Where a0 ,a1 ,a2 ,…,an are given real numbers and x1 ,x2 ,…,xn are independent variables taking all numerical value in the given intervals. Such functions are also called multivariable functions. A multivariable function can be linear and non- liner, for example, Y = 2+3x1 +5x2 (linear multi- variable function) and Y = 3+4x1 +15x1 x2 +10x2 2 (non- linear multivariable function) Multivariable functions may not be graphed easily because these require threedimensional plane or more dimensional plane for plotting the graph. In general, a function with n variables will require (n+1) dimensional plane for plotting its graph. Polynomial Functions: A function of the form Y = f(x) = a1 xn-1 +…+an x0 (1-4) Where a1 ’s(I = 1,2,…,n) are real numbers, a1 0 and n is a positive integer is called a’s polynomial of degree n. a) if n = 1, then the polynomial function is of degree 1 and is called a linear function. That is, for n = 1, function (1-4) cam be written as: y = a1 x1 +an x0 (a1 0) This is usually written as Y = a + bx (since x0 = 1) Where ‘a’ and ‘b’ symbolise an and a1 respectively.

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b) if n = 2, then the polynomial function is of degree 2 and is called a quadratic function. That is, for n = 2, the function (1-4) can be written as: c) y = a1 x2 +a2 x1 +an x0 (a1 0) This is usually written as Y = ax2 +bx+c where a1 = a, a2 = b and an = c Absolute Value Functions The functional relationship expressed by Y = |x| Is known as an absolute value function, where |x| is known as magnitude (or absolute value) of x. By absolute value we mean that whether x is positive or negative, its absolute value remains positive. For example |7|=7 and |-6|=6. Plotting of the graph of the function y=|x|, assigning various values to x and then calculating the corresponding values of y, is shown in the table below:

X Y

… …

-3 3

-2 2

-1 1

0 0

1 1

2 2

3 3

… …

The graph of the given function is shown in Figure 1.5

y y=|- x| 4(-3,3) 3- (3,3) y=|x|

(-2,2) -

2-

-(2,2)

(-1,1) - 1| | |

- (1,1)
| | |

-x

-3

-2

-1

0

1

2

3

+x

Figure 1.5

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13 Inverse Function Take the function y = f(x). Then the value of y, can be uniquely determined for given values of x as per the functional relationship. Sometimes, it is required to consider x as a function of y, so that for given values of y, the value of x can be uniquely determined as per the functional relationship. This is called the inverse function and is also denoted by x=f-1 (y). For example consider the linear function: Y = ax+b Expressing this in terms of x, we get X = y-b/a = y/a-b/a = cy + d where c = 1/a, and d = -b/a This is also a linear function and is denoted by x = f-1 (y) Step Function For different values of an independent variable x in an interval, the dependent variable y=f(x) takes a constant value, but takes different values in different intervals. In such cases the given function y = f(x) is called a step function. For example y1 , if 0 < x 0 and falls to the left a0 as shown in the Figure 1.7 (a) and (b). Figure 1.7(a) Figure 1.7(b)

y y=kax (a>1,k>0) k 0 x k 0 x y=kax (a>1, k>0

b) Logarithmic functions A logarithmic function is expressed as Y=loga x Where a 1 and >0 is the base. It is read as ‘y’ is the log to the base a of x’. this can also be written as X=ay

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15 Thus from an exponential function y=ax , we may construct the logarithmic function x=ay by interchanging the variables. This shows that the inverse of an exponential function is a logarithmic function. The two most widely used bases for logarithms are ’10’ and ‘e’ (=2.7182). i) Common logarithm: It is the logarithm to the base 10 of a number x. it is written as log10 x. if y=log10 x, then x=10y ii) Natural logarithm: It is the logarithm to the base ‘e’ of a number x. it is written as loge x or In x. when no base is mentioned, it will be understood that the base is e. Some important properties of the logarithmic function y=loge x are as follows: i) ii) iii) iv) v) vi) log 1=0 loge=1 log (xy)=log x+log y log (x/y)=log x - log y log (xn ) = n log x loge 10 = 1/ log10 e

vii) loge a = (loge 0) (log10 a) = log 10 a/log10 e viii) logarithm of zero and negative number is not defined. Exercise 1. Draw the graph of the following functions a) y=3x-5 b) y=x2 c) y=log2 x 2. The data of machine operating cost © and the age (t) of the machine are shown in the following table: t (years) : 1 2 3 4 5 c (in ‘000’s) : 5 8 13 20 29 i) Express operating cost as a function of the machine age ii) Sketch the graph of the function derived in (i).

2.4 Solution of Functions
The value (s) of x at which the given function f(x) becomes equal to zero are called the roots (or zeros) of the function f(x). For the linear function Y = ax + b The roots are given by ax + b = 0 Or x = -b/a Thus if x = -b/a is substituted in the given linear function y = ax + b then it becomes equal to zero. In the case of quadratic function, Y = ax2 +bx+c,

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16 We have to solve the equation ax2 +bx+c=0; a 0 to fined the roots of y. The general value of x for which the given quadratic function will become zero is given by -b± (b2-4ac) X = 2a Thus, in general, there are two values of x for which y becomes zero. One value is -b+ (b2-4ac) X= 2a and other value is -b- (b2 -4ac) X= 2a It is very important to note that the number of roots of the given function are always equal to the highest power of the independent variable. Particular Cases: The expression b2 -4ac in the above formula is known as discriminant which determines the nature of the roots as discussed below: i) If b2 -4ac>0, then the two roots are real and unequal ii) If b2 -4ac=0 or b2 =4ac, then the two roots are equal and are equal to – b/2a iii) If b2 -4acp1). The unsold papers at the end of the day are bought by a wastepaper dealer for p3 paise per paper (p3