PSAR Unit 2: Set Theory PDF

PSAR Unit 2: SET THEORY  Set is defined as a well-defined collection of distinct objects. Ex. Set of whole numbers.  A set is generally denoted with a capital alphabet S or A, B, C, etc. and its elements are denoted by small letters a, b, c, etc. generally enclosed in curly brackets like this S = {2, 3 ,4} Here, 2, 3 and 4 are elements of set S and an element is generally denoted using the symbol Є (belongs to) like this: 2 ∈ S  Every object is called Element of the set. Note: 1. There should exist a rule with the help of which we should be in position to tell whether a particular object belongs to that collection or not. 2. Order of elements in a set is not important. 3. The elements of the set may represent quantitative or qualitative characteristics or both. E.g., C = {male, 64 inches} indicate both sex and height of a person.  Methods of Representing Sets: o Tabular form: We write the members of the set, in parenthesis and separate them by putting commas between them. E.g., A = { 2, 4, 6, 8….}  set of even positive nos. o Set-Builder form: This method representing a set is useful when it is not possible to list all possible elements of the set. o When a set is very large and cannot be contained it can also be denoted in the following way: A = {x: x is an integer} or, B = {x: 0 < x < 10, x Є R}, where R is a set of real numbers. Here, we write the properties which all elements of the sets must satisfy and write an element x to represent all elements of the set.  CARDINALITY OF A SET o Cardinality of a set means number of elements in any set. The number of elements in the set is called cardinal number. o For ex, Consider the set above S = {2, 3 ,4}. Now the set contains 3 elements in it and thus its cardinality is 3. o It is denoted by n(S) = 3 or |S| = 3. Types of Sets 1. Null / Empty set: A set with zero or no elements is called Null set. It is denoted by { } or Ø. Null set cardinal number is 0. 2. Singleton set: Sets with only one element in them are called singleton sets. Ex. {2}, {a}, {0}. 3. Finite and infinite set: A set having finite number of elements is called finite set. A set having infinite or uncountable elements in it is called infinite set. E.g., Empty set Ø is a finite set. Set of all positive integers is an infinite set. 4. Universal set: The set of those elements that are of interest in a study or a parent set from which all different subsets are considered is known as a Universal set for that study. i.e., a set which contains all the elements of all the sets and all the other sets in it i.e., subsets of the set, is called Universal set (or fundamental set or master set). 5. Subset: A set is said to be subset of another set if all the elements contained in it are also part of another set. Ex. If A = {1,2}, B = {1,2,3,4} then, set “A” is said to be subset of set B. Symbolically, A C B i.e., if x Є A => x Є B then A C B. Note: 1. If A C B and B C A then A and B are equal sets. i.e., A = B. 2. The empty set Ø is a subset of every set. 3. Every set is subset of itself. 6. Equal sets: Two sets are said to be equal sets if each set is subset of the other. If all elements A are elements of B and all elements of B are elements of A. i.e., when they contain same elements. Symbolically we write, A = B. Ex. A = {a, b, c} and B = {c, b, a} then A and B are called equal sets. Or A = {x: x 2 – 5x + 6 = 0} and B = {2, 3} are equal sets. 7. Disjoint sets: When two sets have no elements in common then, the two sets are called disjoint sets. Ex. A = {1,2,3} and B = {6,8,9} then A and B are disjoint sets. 8. Power set: A power set is defined as the collection of all the subsets of a set and is denoted by P(A). If A = {a, b} then P(A) = { { }, {a}, {b}, {a, b} }. For a set having n elements, the number of subsets in its power set is 2 n. Example 1. List all subsets that can be formed by given set A, where A = {1, 2, 3, 4}. Solution: No of subsets in a power set = 24 = 16. Properties of Sets:  The null set is a subset of all sets.  Every set is subset of itself.  A U (BUC) = (AUB) U C  A ∩ (B∩C) = (A∩B) ∩ C  A U (B∩C) = (AUB) ∩ (AUC)  A ∩ (BUC) = (A∩B) U (A∩C)  AUØ=A Venn diagrams: A Venn diagram is a figure to represent various sets and their relationship, and it is named after English Logician John Venn. The Universal set is represented by a rectangle and subsets of the universal sets are shown through circle, intersecting if they have common elements, non-intersecting if disjoint. If A and B are two Subsets of Universal set U then,  Union of sets is defined as the collection of elements either in A or B or both. It is represented by symbol “U” i.e., A U B A U B = {x / x Є A or x Є B or x Є A and x Є B}.  Intersection of set is the collection of elements which are in both A and B, and is denoted by A ∩ B. i.e., A ∩ B = {x / x Є A and x Є B}.  DIFFERENCE OF SETS: If A and B are two sets then their difference (A – B) is a set containing elements that are only present in A and not in B. Mathematically, A – B = {x: x Є A and x does not belong B} and B – A = {x: x Є B and x ∉ A}. For ex, A = {1, 2} and B = {2, 3, 4}, then A – B = {1}, and B – A = {3,4}  COMPLEMENT OF A SET: Let A be any set then Ac is the set containing all the elements except that are in A i.e., AC = U – A. Example. U = {1, 2, 3, 4}, A = {3, 4}, then A’ = {1, 2}. Note: (A’)’ = A  DISJOINT SETS: A collections of sets are called Disjoint sets if their intersection is empty or i.e., if A and B are two sets then there are no elements common to both A and B. A ∩ B = {} SOME IMPORTANT LAWS OF SET ALGEBRA  COMMUTATIVE LAWS 1. A U B = B U A 2. A ∩ B = B ∩ A  ASSOCIATIVE LAWS 1. (A U B) U C = A U (B U C) 2. (A ∩ B) ∩ C = A ∩ (B ∩ C)  DISTRIBUTIVE LAWS 1. A U (B ∩ C) = (A U B) ∩ (A U C) 2. A ∩ (B U C) = (A ∩ B) U (A ∩ C)  IDENTITY LAWS 1. A U Ø = A, A∩Ø=Ø 2. A ∩ U = A, AUU=U  IDEMPOTENT LAWS : A U A = A, A∩A=A  COMPLEMENT LAWS 1. A U Ac = U 2. A ∩ Ac = Ø 3. (A’)’ = A  LAW OF INCLUSION 1. A C (A U B), B C (A U B) 2. (A ∩ B) C A, (A ∩ B) C B  LAW OF DIFFERENCE 1. A – Ø = A, A–A=Ø  DE MORGAN’S LAW 1. (A U B)C = AC ∩ BC 2. (A ∩ B)C = AC U BC Number of elements in a finite set  Let A and B be two finite sets. Let n(A) and n(B) denote the number of elements in finite sets A and B, respectively. Then, o n(AUB) = n(A) + n(B) - n(A∩B). o n( A U B ) = n(A) + n(B) ( for disjoint sets A and B). o If there are 3 sets A, B and C then, n(AUBUC) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C). o n(A U B U C) = n(A) + n(B) + n(C) ( for three disjoints sets) To maximize overlap,  Union should be as small as possible  Calculate the surplus = n(A) + n(B) + n(C) - n(AUBUC)  This can be attributed to n(A∩B∩C′), n(A∩B′∩C), n(A′∩B∩C), n(A∩B∩C).  To maximize the overlap, set the other three terms to zero. Some other important properties  A’ is called complement of set A, or A’ = U-A  n(A-B) = n(A) - n(A∩B)  A-B = A∩B’  B-A = A’∩B  (A-B) U B = A U B. Cartesian Product of Sets  If a set A has ‘m’ elements and another set has B ‘n’ elements, the m x n ordered pairs such that x ∈ A, y ∈ B, form a set A x B called the Cartesian product of sets A and B, written A x B is expressed as : A x B = {(a,b)|a is every element in A, b is every element in B} Or A x B = {(x,y)|x belongs to A, y belongs to B} NOTE: A x B ≠ B x A For e.g. A = {1,2}, B = {4,5,6}, then, A x B = {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6)} Symmetric Difference of two sets  Let A and B be two sets. The symmetric difference of two sets A and B is the set of elements that belong to only A or only to B and is denoted as A ∆ B. In symbols, A ∆ B = {x / x ∈ A ∩ B’ or x ∈ A’ ∩ B} = (A ∩ B’) U (A’ ∩ B) = (A – B) U (B – A) Note: A – B ≠ B – A e.g., for U = {a, b, c, d}, A = {a, b, c} and B = {b, c, d} verify that (A U B) – B = A ∩ B’ Venn diagram in case of two Sets Where; X = number of elements that belong to set A only Y = number of elements that belong to set B only Z = number of elements that belong to set A and B both (AB) W = number of elements that belong to none of the sets A or B From the above figure, it is clear that n(A) = x + z ; n (B) = y + z ; n(A ∩ B) = z; n ( A ∪ B) = x +y+ z. Total number of elements = x + y + z + w. In case of three sets, always start from innermost part and go outwards to get the exact number of elements in each set. EXAMPLES: 1. In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both tea and coffee. 1. How many students like only tea? 2. How many students like only coffee? 3. How many students like neither tea nor coffee? 4. How many students like only one of tea or coffee? 5. How many students like at least one of the beverages? Solution: The given information may be represented by the following Venn diagram, where T = tea and C = coffee. o Number of students who like only tea = 60 o Number of students who like only coffee = 40 o Number of students who like neither tea nor coffee = 200 – (60 +40 +80) =200 – 180 = 20 o Number of students who like only one of tea or coffee = 60 + 40 = 100 o Number of students who like at least one of tea or coffee = n(only Tea) + n(only coffee) + n(both Tea & coffee) = 60 + 40 + 80 = 180 2. In a survey of 500 students of a college, it was found that 49% liked watching football, 53% liked watching hockey and 62% liked watching basketball. Also, 27% liked watching football and hockey both, 29% liked watching basketball and hockey both and 28% liked watching football and basketball both. 5% liked watching none of these games.  How many students like watching all the three games?  Find the ratio of number of students who like watching only football to those who like watching only hockey.  Find the number of students who like watching only one of the three given games.  Find the number of students who like watching at least two of the given games. Solution: n(F) = percentage of students who like watching football = 49% n(H) = percentage of students who like watching hockey = 53% n(B)= percentage of students who like watching basketball = 62% n ( F ∩ H) = 27% ; n (B ∩ H) = 29% ; n(F ∩ B) = 28% Since 5% like watching none of the given games so, n (F ∪ H ∪ B) = 95%. Now applying the basic formula, 95% = 49% + 53% + 62% -27% - 29% - 28% + n (F ∩ H ∩ B) Solving, you get n (F ∩ H ∩ B) = 15% Now, make the Venn diagram as per the information given Note: All values in the Venn diagram are in percentage  Number of students who like watching all the three games = 15 % of 500 = 75.  Ratio of the number of students who like only football to those who like only hockey = (9% of 500)/(12% of 500) = 9/12 = 3:4.  The number of students who like watching only one of the three given games = (9% + 12% + 20%) of 500 = 205  The number of students who like watching at least two of the given games = (number of students who like watching only two of the games) +(number of students who like watching all the three games) = (12 + 13 + 14 + 15) % i.e. 54% of 500 = 270 3. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French? Let A denote people who can speak English Let B denote people who can speak French Given that: n(A U B) = 100 To find: 1. People who can speak English only i.e., only A 2. People who can speak French only i.e., only B 3. People who can speak both English and French only i.e. A ∩ B Solution: Since n(A U B) = n(A) + n(B) – n(A ∩ B) n(A ∩ B) = 72 + 43 – 100 = 15 Now only A can be written as = n(A) – n(A ∩ B) = 72 – 15 = 57 Similarly, only B = n(B) – n(A ∩ B) = 43 -15 = 28 4. Of the 200 candidates who were interviewed for a position at a call centre, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone. 40 of them had both, a two-wheeler, and a credit card, 30 had both, a credit card, and a mobile phone and 60 had both, a two-wheeler and mobile phone and 10 had all three. How many candidates had none of the three? Solution: Let A denote candidates who have 2-wheelers. Let B denote candidates with credit card. Let C denote candidates with mobile phones. Now we need to find n(Ac∩ Bc ∩ Cc) This can easily be calculated using De’ Morgan’s Law, i.e. n (A U B U C)c = n(Ac∩ Bc ∩ Cc) Since, n(A U B U C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) = 100 +70 +140 – 40 – 30 – 60 + 10 = 190 Total Candidates are 200, therefore, n(A U B U C) + n (A U B U C)c =200 n (A U B U C)c = 200 – 190 = 10 5. In a class 40% of the students enrolled for Math and 70% enrolled for Economics. If 15% of the students enrolled for both Math and Economics, what % of the students of the class did not enrol for either of the two subjects? A. 5% B. 15% C. 0% D. 25% E. None of these Correct Answer A. 5% Sol: n(A ∪ B) = n(A) + n(B) - n(A ∩ B), where (A ∪ B) represents the set of people who have enrolled for at least one of the two subjects Math or Economics and (A ∩ B) represents the set of people who have enrolled for both the subjects Math and Economics. n(A ∪ B) = 40 + 70 - 15 = 95% That is 95% of the students have enrolled for at least one of the two subjects Math or Economics. Therefore, the balance (100 - 95)% = 5% of the students have not enrolled for either of the two subjects. 6. In a city whose population is 60000, there 35000 people read Hindi newspaper, 25000 people read English newspaper, and 8000 people read Hindi and English newspaper then how many people do not read any newspaper? 1. A) 9000 B) 10000 2. C) 8000 D) 12000 Answer:- Total people who read newspaper 35000 + 25000 – 8000 = 52000 So, numbers of people who do not read news paper = 60000 – 52000 = 8000 So, option number (C) is right. For Maximum and Minimum of values, the key point to note is: If you allot a value to the intersection, it will get added to all the individual sets but will bring down the total. 7. In a survey it was found that 80% like tea whereas 70% like coffee. What is the maximum and minimum number of those who like both? Sol: First thing to note is that no information is mentioned about the people who don’t like either of them. So that value is flexible and can change. n(tea) = 80 n(coffee) = 70 n(total) = 100 {This includes those who like neither.} n(tea n coffee) = ??? {We don’t know this value and it is flexible} If we want to maximize those who like both, we have to maximize the value in the intersection. So, we have to minimize the value of the union. n(tea n coffee)max = 70 {It is limited by the higher of the two values} In this case 20% of people like neither tea nor coffee. If we want to minimize those who like both, we have to minimize the value in the intersection. So, we have to maximize the value of the union. We know that the maximum possible value of the union ie, n(tea U coffee) = 100 So, we need to figure out the surplus : n(tea) + n(coffee) = 80 + 70 = 150. The surplus is = 150 – 100 = 50 So, the value of the intersection = value of the surplus = 50 This could have also been obtained by the formula n(a U b) = n(a) + n(b) – n (a n b). In this case, there is no one who likes neither coffee nor tea. MCQs 1. Let A and B be two sets. Write the expression for subsets for questions mentioned below, in the context of A and B. a) Elements belong to at least one of A or B. (A U B) b) Elements belong to both sets A and B. (A ∩ B) c) Elements belong to neither set A nor set B d) Belongs to A but not in B e) Belongs to exactly one of set A or B. f) Not more than one of the sets A or B g) Belongs to A, so belongs to B 2. Let A, B, C are three sets, Write the expression for the following subset noted below, in the context of sets A, B, C a) Belongs to only A b) Belongs to all three c) Belongs to at least two d) Belongs to two and no more sets e) Belongs to both A and B but not in C f) Belongs to at least one g) Does not belong to any three sets A, B, C 3. If S ={ 1,2,3,4,5,6,7,8,9}. A = {1,3,5,7} B= {6,7,8,9} C = {2,4,8} and D = {1,5,9}. 1. The elements of the set A´ ∩ B is (a) {1,2,3,4} (b) {2,4,6,8,9} (c) {6,8,9} (d) none 2. The elements of the set (A´ ∩ B) ∩ C (a) {8} (b) {6} (c) {6,8,9} (d) none 3. The elements of set B´ ∪ C (a) {2,4} (b) {1,2,3,4,5,8} (c) {1,2,3} (d) none 4. The elements of set (B´ ∪ C) ∩ D (a) {1,5} (b) {φ} (c) φ (d) {1,2,4,9} 5. The elements of set A´ ∩ C (a) Set C (b) {6,9} (c) {1,2,3} (d) none 6. Let A = { 1,2,3,4} and B={H,T} find A × B. (a) {1,2,3,4,H,T} (b) {(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H),(4,T)} (c) φ (d) None 7. Number of subsets of set C = {Profit, loss, breakeven point} (a) 8 (b) 3 (c) zero (d) none 8. If V ={ x:x+2 = 0} , R = {x: x 2 + 2x = 0} and S= {x : x 2 + x -2 =0} then value of x for which V,R,S are equal. (a) 2 (b) -2 (c) 0 (d) none A sample of income of 1172 families was surveyed and noticed that for income groups less than Rs. 6000/- , Rs.6000/- to Rs.10999/-, Rs.11000/- to Rs. 15999/- and RS. 16000/- or more: no TV set is available to 70, 50,20,50; one TV set available to 152,308,114,46 families ; two or more than two TV sets available to 10,174,84, 94 families. If A= { x| x is a family owning two or more sets} , B = {x |x is a family with one set}, C = {x |x is a family with income less than Rs.6000/-}, D ={x |x is a family with incomeRs.6000/- to Rs.10999/-} E ={x |x is a family with incomeRs.11000/- to Rs.15999/-} Answer Q-9,10,11 9. The number of families in the sets (C ∩B ) (a) 152 (b) 468 (c) 174 (d) none 10. The number of families in the sets A∪E (a) 308 (b) 496 (c) 982 (d) none 11. Find the number of families in each of the following sets :— (i) (A∪B)’∩E (ii) (C∪D∪E)∩(A∪B)’ (a) 20,50 (b) 152,20 (c) 152,50 (d) 20,140 12. De Morgan’s Law state (a) (A∪B) =(A∩ B) (b) (A∪B)’ =(A’∩ B’) (c) (A∪B)’ =(A’∪ B’) (d) None 13. Number of subsets of a set having three elements are (a) 8 (b) 3 (c) 12 (d) none 14. If S= { x/0<x<10, X is integer}, M = {x/3<x≤8, X is integer ) and N={x/5<x<10 ,X is integer}, then elements of M’∪N are (a) null s0065t (b) {1, 2,3,4} (c) {1,2,3,6,7,8,9} (d)none 15. In a group of 200 college students 138 are enrolled in a course of psychology, 115 are enrolled in a course in sociology and 91 are enrolled in both. How many of these students are not enrolled in either course? (a) 38 (b) 47 (c) 24 (d) none 16. A market research organization claims that among 500 shoppers interviewed, 308 regularly buy product X, 266 regularly buy product Y, 103 regularly buy both, and 59 buys either on regular basis. Using Venn diagram and filling in the number of shoppers associated with the various regions, check whether the data are consistent? (a) no (b) Yes (c) cannot say anything (d) Sometimes consistent Among 120 visitors to Disneyland, 74 stayed for at least 3 hours, 86 spent at least $20, 64 went on the Matterhorn ride, 60 stayed for at least 3 hours and spent at least $20, 52 stayed for at least 3 hours and went on the Matterhorn ride, 54 spent at least $20 and went on Matterhorn ride, and 48 stayed for at least 3 hours, spent at least $20 and went on Matterhorn ride. Drawing a Venn diagram, Q-16,17,18 17. How many of the 120 visitors to Disneyland stayed for at least 3 hours, spent at least $20, but did not go on the Matterhorn ride? (a) 12 (b) 6 (c) 48 (d) none 18. How many of the 120 visitors went on the Matterhorn ride, but stayed less than 3 hours and spent less than $20? (a) 12 (b) 6 (c) 64 (d)none 19. How many of the 120 visitors stayed less than 3 hours, spent at least $20, but did not go on the Matterhorn ride. (a) 48 (b) 86 (c) 20 (d) none 20. A company manufactures soap, powder and toothpaste and wishes to know about its market in a particular area. A survey was conducted covering a population of 1899 persons. From the survey report the manager came to know that 30 persons had been using all the 3 products, 757 persons had been using soap, 574 persons powder, 132 persons soap and powder, 155 persons soap and toothpaste, 472 persons toothpaste and 47 persons toothpaste and powder. Also, it was stated that 398 persons did not use any of their products. The data represents (a) consistency (b) inconsistency (c) dependency (d) independency. In a survey of 100 students in a music school, the number of students learning different musical instruments were found to be guitar 28; Venn 30;flute 42; guitar and Venn 8; guitar and flute 10 ; Venn and flute 5; all musical instruments 3. Answer Q-20,21 21. How many students were learning none of these three musical instruments? (a) 12 (b) 20 (c) 42 (d) none 22. How many students were learning only the flute? (a) 30 (b) 42 (c) zero (d)none. 23. A manufactured item can have a defect in its shape or in its weight. A sample of 100 items showed that 10 of them had defect in its shape and 12 of them in weight. Five of them had both the defects. How many of them had exactly one defect? (a) 10 (b) 12 (c) 50 (d) none Krishna Iron and Steel Company produces iron rods of length 5 feet, radius 2 inches and weight 400 kg. After the production, the quality controller measures the above to check if they conform to standards. On a certain day there were 14 rods classified as defective by the quality controller. The causes of defect were classified as follows: Defects No. of rods Weight 7 Radius 5 Length 8 Also, there were 2 rods with defects in weight and length, 3 rods with defects in length and radius; and 1 in weight and radius; and 1 with all three defects. Using Venn diagram AnswerQ-24, 25, 26. 24. How many rods were there with defect in Weight alone? (a) 3 (b) 5 (c) 2 (d) none 25. How many rods were there with defect in Radius alone? (a) 1 (b) 2 (c) zero (d) none 26. How many rods were there with defect in Length alone? (a) 1 (b) 2 (c) 4 (d) none A survey was conducted in a city to study the preference for three newspapers: Hindustan Herald (H), Bharat News (B) and India Tribune (T). The total numbers of persons interviewed were 200. 92 people said that they read Hindustan Herald, 86 read Bharat news, and 83 read India Tribune. Also 25 of them read Hindustan Herald and Bharat News, 27 read Hindustan Herald and India Tribune, 26 read Bharat News and India Tribune and 9 read all the three papers. Answer the Que- 27 to 34. 27. How many elements are there in set H only (a)23 (b) 22 (c) 49 (d) none 28. How many elements are there in set B only (a) 92 (b) 44 (c) 24 (d) none 29. How many elements are there in set H∪B (a) 153 (b) 23 (c) 27 (d) none 30. How many elements are there in set H∩B (a) 83 (b) 23 (c) 25 (d) none 31. How many elements are there in set T∩(H∪B) (a) 24 (b) 20 (c) 44 (d) none 32. How many elements are there in set of Readers of exactly two newspapers (a) 104 (b) 198 (c) 51 (d) none 33. How many elements are there in set of Readers of exactly one newspaper. (a) 132 (b) 100 (c) 230 (d) none 34. How many elements are there in set of readers of no newspapers (a) 24 (b)8 (c) 34 (d) none. 35. The dimension of the element of the set of Cartesian products of three set A , B and C is (a) 27 (b) 3 (c) one (d) none 36. Cardinality of set A is denoted by (a) |A| (B) n(A) (c) A (d) none 37. The number of elements of the set of Cartesian products of sets A: {a,b,c} ,B:{H,T} and C:{1,2,3,4} is (a)24 (b) 9 (c) 3 (d) none. 38. Following set notations represents :- A⊂B; x ∉ A; A⊃B; {0}; A⊄B (a) A is proper subset of B; x is not an element of A; A contain B; singleton set with an only element zero; A is not contained in B (b) A is proper subset of B; x is an element of A; A contain B; singleton set with an only element zero; A is not contained in B (c) A is proper subset of B; x is not element an of A; A does not contain B; contain elements other than zero ; A is not contained in B (d) None of them. 39. If V ={0,1,2…,9}, X={0,2,4,6,8} Y={3,5,7} and Z={3,7} then Y∪Z, (V∪Y)∩X, (X∪Z)∪V are respectively:⎯ (a) {3,5,7}, {0,2,4,6,8}, {0,1,2…,9} (b) {2,4,6}, {0,2,4,6,8}, {0,1,2…,9} (c) {2,4,6}, {0,1,2…,9}, {0,2,4,6,8}, (d) None 40. In Question No (38 ) (X∪Y)∩Z and (φ ∪ V)∩φ are respectively:⎯ (a) {0,2,4,6,8},φ (b) {3,7}, φ (c) {3,5,7}, φ (d) None 41. If V = { x : X+2 <0 } R= {X: X 2 +X -2 =0, X<0 } and S={ X: X 2 +4X+4 =0} then V, R, S are equal, for the value of X equal to _____. (a) 0 (b) -1 (c) -2 (d) none 42. What is the relationship between the following sets? A = {X: X is a letter in the word flower} B= {X: X is a letter in the word flow} , C = {X: X is a letter in the word wolf } , D = {X: X is a letter in the word follow}, (a) B = C = D and all these are subsets of the set A (b) B = C ≠ D (c) B ≠ C ≠ D (d) none. 43. Comment on the correctness or otherwise of the following statements:⎯ (i) {a, b, c}={c, b, a} (ii) {a, c, a, d, c, d} ⊂ {a, c, d} (iii) {b} ∈[{b}} (iv) {b} ⊂ {{b}} and φ⊂{{b}} (a) Only (iv) is incorrect (b) (i) and (ii) are incorrect (c) (ii) and (iii) are incorrect (d) all are incorrect. 44. If A = {a, b, c}, B= {a, b} , C={a, b, d} , D= {c, d} and E ={d} state which of the following statements are correct :⎯ (i) B ⊂ A (ii) D ≠ C (iii) C ⊃ E (iv) D ⊂ E (v) D ⊂ B (vi) D= A (vii) B ⊄ C (viii) E ⊂ A (ix) E ⊄ B (x) a ∈A (xi) a ⊂A (xii) {a} ∈A (xiii) {a} ⊂ A (a) (i), ( ii ), (iii), ( ix), (x), (xiii) only are correct (b) (ii), (iii), ( iv), ( x), (xii), (xiii) only are correct (c) (i), (ii), (,iv), (ix), (xi), (xiii) only are correct (d) none. 45. Let A = {0}, B = {0,1}, C = φ, D = {φ}, E = x; x is a human being 300 years old}, F = {x; x ∉ A and x ∉ B } , State which of the following statements are true:— (i) A⊂B (ii) B = F (iii) C ⊂ D (iv) C = E (v) A =F (vi) F = 1 and (vii) E = C = D. (a) (i) (iii) (iv) and (v) only are true. (b) (i) (ii) (iii) and (iv) are true (c) (i) (ii) (iii) and (vi) only are true (d) none 46. If A = {0,1} state which of the following statements are true:— (i) {1} ⊂ A (ii) {1} ∈A (iii) φ ∈ A (iv) 0∈A (v) 1⊂ A (vi) {0}∈ A (vii) φ⊂ A (a) (i) (iv) and (vii) only are true (b) (i) (iv) and (vi) only are true (c) (ii) (iii) and (vi) only are true. (d) none 47. State whether the following sets are finite, infinite, or empty:— (i) X = {1,2,3…,500} (ii) Y = { y: y = a2 ; a is an integer } (iii) A = {x: x is a positive integer multiple of 2} (iv) B = {x : x is an integer which is a perfect root of 26 < x < 35} (a) finite, infinite, infinite, empty (b) infinite, infinite, finite, empty. (c) infinite, finite, infinite, empty (d) none. 48. If A = { 1,2,3,4} B = {2,3,7,9} and C = {1,4,7,9} then (a) A ∩ B ≠ φ B∩ C ≠ φ A ∩ C ≠ φ but A ∩ B ∩ C = φ (b) A ∩B = φ B ∩ C = φ A ∩ C = φ A ∩ B ∩ C = φ (c) A ∩ B ≠φ B∩C ≠ φ A ∩ C ≠ φ A ∩ B ∩ C ≠ φ (d) none 49. If the universal set is X = { x: x ∈ N ; 1≤x≤12} and A = {1,9,10} , B = {3,4,6,11,12} and C =(2,5,6} are subsets of X then set A∪(B∩C) is ______ (a) {3,4,6,12} (b) {1,6,9,10} (c) {2,5,6,11} (d) none. 50. As per Question no. (48) the set (A∪B)∩(A∪C) is ______. (a) { 3,4,6,12} (b) {1,6,9,10} (c) {2,5,6,11} (d) none Let P = { 1,2,x} Q = {a, x, y} R ={ x, y, z } then 51.P×Q is 52. P×R 53. Q×R 54. (P×Q)∩(P×R) is (a) {(1,a) (1,x) (1,Y) (2,a) (2,x) (2,y) (x, a) (x, x) (x, y)} (b) {(1,x) (1,y) (1,z) (2,x) (2,y) (2,z),(x, x) (x, y) (x, z)} (c) {(a, x) (a, y) (a, z) (x, x) (x, y) (x, z) (y, x) (y, y) (y, z) } (d) {(1,x) (1,y) (2,x) (2,y) (x, x) (x, y) 55. Identify the elements of set P if set Q = {1,2,3} and P×Q= {(4,1) (4,2) (4,3) (5,1) (5,2) (5,3) (6,1) (6,2) (6,3) } (a) {3,4,5} (b) {4,5,6} (c) {5,6,7} (d) none 56. Complaints about works canteen had been about Mess (M) , Food (F) and Service(S) total complaints 173 were received as follows:— n(M) = 110, n(F) = 55, n(S) = 67, n(M∩F∩S’) = 20 , n(M∩S∩F’) = 11 and n(F∩S∩M’)= 16. Determine the complaints about all the three. (a) 6 (b) 53 (c) 35 (d) none ------------------------- ANSWERS TO MCQ’S OF SET THEORY:1c 2a 3b 4a 5a 6b 7a 8b 9a 10b 11d 12b 13a 14c 15a 16a 17a 18b 19c 20b 21b 22a 23b 24b 25b 26c 27c 28b 29a 30c 31c 32c 33a 34b 35b 36b 37a 38a 39a 40b 41c 42a 43c 44a 45d 46a 47a 48a 49b 50b 51a 52b 53c 54d 55b 56a

PSAR Unit 2: Set Theory PDF

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