Chapter 7 Sets, Relations and Functions, Basics of Limits and Continuity functions PDF

CHAPTER 7 SETS, RELATIONS AND FUNCTIONS After reading this chapter, students will be able to understand:  Understand the concept of set theory.  Appreciate the basics of functions and relations.  Understand the types of functions and relations.  Solve problems relati...

CHAPTER 7 SETS, RELATIONS AND FUNCTIONS After reading this chapter, students will be able to understand:  Understand the concept of set theory.  Appreciate the basics of functions and relations.  Understand the types of functions and relations.  Solve problems relating to sets, functions and relations. CHAPTER OVERVIEW The Concept of Set Theory Subset Types of Sets Relations Functions Types of Types of Relations Functions In our mathematical language, everything in this universe, whether living or non-living, is called an object. If we consider a collection of objects given in such a way that it is possible to tell beyond doubt whether a given object is in the collection under consideration or not, then such a collection of objects is called a well-defined collection of objects. © The Institute of Chartered Accountants of India 7.2 BUSINESS MATHEMATICS A set is defined to be a collection of well-defined distinct objects. This collection may be listed or described. Each object is called an element of the set. We usually denote sets by capital letters and their elements by small letters. Example: A = {a, e, i, o, u} B = {2, 4, 6, 8, 10} C = {pqr, prq, qrp, rqp, qpr, rpq} D = {1, 3, 5, 7, 9} E = {1,2} etc. This form is called Roster or Braces form. In this form we make a list of the elements of the set and put it within braces { }. Instead of listing we could describe them as follows : A = the set of vowels in the alphabet B = The set of even numbers between 2 and 10 both inclusive. C = The set of all possible arrangements of the letters p, q and r D = The set of odd digits between 1 and 9 both inclusive. E = The set of roots of the equation x2 – 3x + 2 = 0 Set B, D and E can also be described respectively as B = {x : x = 2m and m being an integer lying in the interval 0 < m < 6} D = {2x – 1 : 0 < x < 5 and x is an integer} E = {x : x2 – 3x + 2 = 0} This form is called set-Builder or Algebraic form or Rule Method. This method of writing the set is called Property method. The symbol : or/reads 'such that'. In this method, we list the property or properties satisfied by the elements of the set. We write, {x:x satisfies properties P}. This means, "the set of all those x such that x satisfies the properties P". A set may contain either a finite or an infinite number of members or elements. When the number of members is very large or infinite it is obviously impractical or impossible to list them all. In such case. we may write as : N = The set of natural numbers = {1, 2, 3…..} W = The set of whole numbers = {0, 1, 2, 3,…) etc. © The Institute of Chartered Accountants of India  SETS, RELATIONS AND FUNCTIONS 7.3 I. The members of a set are usually called elements. In A = {a, e, i, o, u}, a is an element and we write a  A i.e. a belongs to A. But 3 is not an element of B = {2, 4, 6, 8, 10} and we write 3B. i.e. 3 does not belong to B. II. If every element of a set P is also an element of set Q we say that P is a subset of Q. We write P  Q. Q is said to be a superset of P. For example {a, b}  {a, b, c}, {2, 4, 6, 8, 10}  N. If there exists even a single element in A, which is not in B then A is not a subset of B. III. If P is a subset of Q but P is not equal to Q then P is called a proper subset of Q. IV.  has no proper subset. Illustration: {3} is a proper subset of {2, 3, 5}. But {1, 2} is not a subset of {2, 3, 5}. Thus if P = {1, 2} and Q = {1, 2 ,3} then P is a subset of Q but P is not equal to Q. So, P is a proper subset of Q. To give completeness to the idea of a subset, we include the set itself and the empty set. The empty set is one which contains no element. The empty set is also known as null or void set usually denoted by { } or Greek letter  , to be read as phi. For example the set of prime numbers between 32 and 36 is a null set. The subsets of {1, 2, 3} include {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and { }. A set containing n elements has 2n subsets. Thus a set containing 3 elements has 23 (=8) subsets. A set containing n elements has 2n –1 proper subsets. Thus a set containing 3 elements has 23 – 1 =7 subsets. The proper subsets of { 1,2,3} include {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3], { }. Suppose we have two sets A and B. The intersection of these sets, written as A  B contains those elements which are in A and are also in B. For example A = {2, 3, 6, 10, 15}, B = {3, 6, 15, 18, 21, 24} and C = { 2, 5, 7}, we have A  B = { 3, 6, 15}, A  C = {2}, B  C =  , where the intersection of B and C is empty set. So, we say B and C are disjoint sets since they have no common element. Otherwise sets are called overlapping or intersecting sets. The union of two sets, A and B, written as A  B contain all these elements which are in either A or B or both. So A  B = {2, 3, 6, 10, 15, 18, 21, 24} A  C = {2, 3, 5, 6, 7, 10, 15} A set which has at least one element is called non-empty set. Thus the set { 0 } is non-empty set. It has one element say 0. Singleton Set: A set containing one element is called Singleton Set. For example {1} is a singleton set, whose only element is 1. Equal Set: Two sets A & B are said to be equal, written as A = B if every element of A is in B and every element of B is in A. Illustration: If A = {2, 4, 6} and B = {2, 4, 6} then A = B. Remarks : (I) The elements of the two sets may be listed in any order. © The Institute of Chartered Accountants of India 7.4 BUSINESS MATHEMATICS Thus, {1, 2, 3} = {2, 1, 3} = {3, 2, 1} etc. (II) The repetition of elements in a set is meaningless. Example: {x : x is a letter in the word "follow"} = {f, o, l, w} Example: Show that  , {0} and 0 are all different. Solution:  is a set containing no element at all; {0} is a set containing one element, namely 0. And 0 is a number, not a set. Hence  ,{0} and 0 are all different. The set which contains all the elements under consideration in a particular problem is called the universal set denoted by S. Suppose that P is a subset of S. Then the complement of P, written as Pc (or P') contains all the elements in S but not in P. This can also be written as S – P or S ~ P. S – P = {x : x  S, x  P}. For example let S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} P = {0, 2, 4, 6, 8} Q = {1, 2, 3, 4, 5) Then P' = {1, 3 ,5 ,7, 9} and Q'= {0 , 6 , 7, 8, 9} Also P  Q = {0, 1, 2, 3, 4, 5, 6, 8}, (P  Q)' = {7, 9} P  Q = {2, 4} P  Q' = {0, 2, 4, 6, 7, 8, 9}, (P  Q)' = {0, 1, 3, 5, 6, 7, 8, 9} P'  Q' = {0, 1, 3, 5, 6, 7, 8, 9} P'  Q' = {7, 9} So it can be noted that (P  Q)' = P'  Q' and (P  Q)' = P'  Q'. These are known as De Morgan’s laws. We may represent the above operations on sets by means of Euler - Venn diagrams. S P P Q (a) Fig. 1 (b) © The Institute of Chartered Accountants of India  SETS, RELATIONS AND FUNCTIONS 7.5 Thus Fig. 1(a) shows a universal set S represented by a rectangular region and one of its subsets P represented by a circular shaded region. The un-shaded region inside the rectangle represents P'. Figure 1(b) shows two sets P and Q represented by two intersecting circular regions. The total shaded area represents PUQ, the cross-hatched "intersection" represents P  Q. The number of distinct elements contained in a finite set A is called its cardinal number. It is denoted by n(A). For example, the number of elements in the set R = {2, 3, 5, 7} is denoted by n(R). This number is called the cardinal number of the set R. Thus n(AUB) = n(A) + n(B) – n(A  B) If A and B are disjoint sets, then n(AUB) = n(A) + n(B) as n (A  B) = 0 A B For three sets P, Q and R n(PUQUR) = n(P) + n(Q) + n(R) – n(P  Q) – n(Q  R) – n(P  R) + n(P  Q  R) When P, Q and R are disjoint sets = n(P) + n(Q) + n(R) Illustration: If A = {2, 3, 5, 7} , then n(A) = 4 Equivalent Set: Two finite sets A & B are said to be equivalent if n (A) = n(B). Clearly, equal sets are equivalent but equivalent sets need not be equal. © The Institute of Chartered Accountants of India 7.6 BUSINESS MATHEMATICS Illustration: The sets A = {1, 3, 5} and B = {2, 4, 6} are equivalent but not equal. Here n (A) = 3 = n (B) so they are equivalent sets. But the elements of A are not in B. Hence they are not equal though they are equivalent. Power Set : The collection of all possible subsets of a given set A is called the power set of A, to be denoted by P(A). Illustration: (i) If A = {1, 2, 3} then P(A) = { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3},  } (ii) If A = {1, {2}}, we may write A = {1, B} when B = {2} then P(A) = {  , {1}, {B }, {1, B}} = {  , {1}, {{2}}, {1,{2}}} Choose the most appropriate option or options (a) (b) (c) or (d). 1. The number of subsets of the set {2, 3, 5} is (a) 3 (b) 8 (c) 6 (d) none of these 2. The number of subsets of a set containing n elements is (a) 2n (b) 2–n (c) n (d) none of these 3. The null set is represented by (a){  } (b) { 0 } (c)  (d) none of these 4. A = {2, 3, 5, 7} , B = { 4, 6, 8, 10} then A  B can be written as (a) { } (b) {  } (c) (AUB)' (d) None of these 5. The set {x|0 0 = Right-hand limit = xlim = f (x) lim 2x 0 →0 + x →0 + Since L.H. limit = R.H. Limit, the limit exists. Thus, lim f(x) = 0. x →0 2 x + 4x + 3 Example 3: lim. x →3 x 2 + 6x + 9 2 2 Solution: x + 4x + 3 = x + 3x + x + 3 = x(x + 3) + 1(x + 3) = (x + 3)(x + 1) = x + 1 x 2 + 6x + 9 ( x + 3)2 (x + 3) (x + 3) 2 2 x+3 x 2 + 4x + 3 x +1 4 2 ∴ lim 2 = lim ==. x → 3 x + 6x + 9 x →3 x + 3 3 3 Example 4: Find the following limits: x −3 x+h − x (i) lim (ii) lim if h > 0. x →9 x−9 h →0 h Solution: x −3 x −3 1 x −3 1 1 = =. ∴ lim = lim =. (i) x−9 ( x +3 )( x −3 ) x +3 x →9 x−9 x →9 x + 3 6 x+h− x x+h-x 1 x+h − x 1 = = ∴ lim = lim (ii) h h x+h + x ( x+h + x h →0 h ) h →0 x+h + x 1 1 1 = = = lim x+h + lim x x + x 2 x. h →0 h →0 3x+ x Example 5: Find lim. x →0 7x-5 x 3x+ x 3x + x Solution: Right-hand limit = lim = lim = lim 2 = 2 x →0 + 7x-5 x x → 0 + 7x − 5x x →0 + 3x+ x 3x-(x) 1 1 Left-hand limit = lim = lim = lim =. x →0 - 7x-5 x x →0 7x − 5 ( −x ) x →0 - 6 6 Since Right-hand limit ≠ Left-hand limit the limit does not exist. © The Institute of Chartered Accountants of India 7.28 BUSINESS MATHEMATICS ex - e-x Example 6: Evaluate = lim x →0 x ex - e-x (ex -1)- (e-x − 1) ex - 1 ex - 1 Solution: = lim == lim = lim − = lim = 1−1= 0 x →0 x x →0 x x →0 x x →0 x x  9 Example 7: Find lim  1+ . (Form 1) x →á  x x x Solution: It may be noted that approaches α as x approaches ∞. i.e. lim → ∞ 9 x →∞ 9 9    x/9  9 x  1   lim  1 + = lim  1 +   x →∞  x  x/9→α  x   9   9  1  z Substitution x / 9 = z the above expression takes the form lim  1 +   z →α   z   9   1   z = lim  1 +   = e9.  z →∞  z   2x +1  ∞ Example 8: Evaluate: lim. Form ∞  z →∞ x3 + 1 Solution: As x approaches ∞ , 2x + 1 and x 3 + 1 both approach ∞ and therefore the given function ∞ takes the form which is determinate. Therefore, instead of evaluating directly let us try for ∞ suitable algebraic transformation so that the indeterminate for is avoided.  2+ 1  2 1 2 1 lim  x 2 x 3  + x →∞   lim x lim x + 2 3 x2 x3   = x →∞ x →∞ 0+0 0 lim 1 = 1 = = = 0 x →∞  1  1+ 0 1 1+ x 3 lim  1 + x  lim1 + lim x 3 3 x →∞   x →∞ x →∞   12 + 22 + 3 2 +.............. + x 2 Example 9: Find lim x →∞ x3 lim [x(x +)(2x + 1)=] 1  1  1  lim  1 +  2 +   3 x →∞ 6x 6 x →∞  x  x  1 1 = x1x2=. 6 3 © The Institute of Chartered Accountants of India  SETS, RELATIONS AND FUNCTIONS 7.29 7.9 CONTINUITY: By the term continuous we mean some thing which goes on without interruption and without abrupt changes. Here in mathematics the term continuous carries the same meaning. Thus, we define continuity of a function in the following way. A function f(x) is said to be continuous at x= a if and only if (i) f(x) is defined x= a (ii) lim f ( x ) = lim f ( x) x  a− x  a+ (iii) lim f ( x ) = f ( a) xa In the second condition both left-hand and right-hand limits are equal. In the third condition limiting value of the function must be equal to its function value at x= a Useful Information: (i) The sum, difference and product of two continuous functions is a continuous function. This property holds good for any finite number of functions. (ii) The quotient of two continuous functions is continuous function provided the denominator is not equal to zero. Example:1 f(x) = [ 3 1 −x when