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### MB1101 QR

1. 1. In this document you can review important • Definitions • Facts • Formulae • Procedures Class – XI, CBSE MB1101: Sets Learn Math for Free. Visit www.ganitgurooz.com
2. 2. What is a Set? Class – XI MB1101: Sets Important Definitions Topic: Sets Learn Math for Free. Visit www.ganitgurooz.com
3. 3. What is a Set? Class - XI MB1101: Sets Important Definitions Topic: Sets A set is a "well defined collection of objects, where the order in which the elements are listed is immaterial and while listing the elements repetition is not allowed. Learn Math for Free. Visit www.ganitgurooz.com
4. 4. What are the symbols for some important sets? Class - XI MB1101: Sets Important Definitions Topic: Sets Learn Math for Free. Visit www.ganitgurooz.com
5. 5. What are the symbols for some important sets? Class - XI MB1101: Sets Important Definitions Topic: Sets N: the set of all natural numbers Z: the set of all integers Q: the set of all rational numbers R: the set of real numbers Z+ : the set of positive integers Q+ : the set of positive rational numbers R+ : the set of positive real numbers Learn Math for Free. Visit www.ganitgurooz.com
6. 6. What is a Notation? Class - XI MB1101: Sets Important Definitions Topic: Sets Learn Math for Free. Visit www.ganitgurooz.com
7. 7. What is a Notation? Class - XI MB1101: Sets Important Definitions Topic: Sets If is an element of a set , we say that " belongs to " the Greek symbol (epsilon) is used to denote the phrase ' '. Thus, we write . . If ' ' is not an element of a set , we write a A a A belongs to a b b A b A and read " does not belong to ". Thus, in the set = 1, 2, 3 , 1 but 4 . b A A A A Learn Math for Free. Visit www.ganitgurooz.com
8. 8. How to represent a set in Roster or Tabular Form? Class - XI MB1101: Sets Important Definitions Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com
9. 9. How to represent a set in Roster or Tabular Form? Class - XI MB1101: Sets Important Definitions Topic: Representations of Sets In roster form, all the elements of a set are listed, separated by commas and are enclosed within braces { }. For example, the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}. The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots tell us that the list of odd numbers continue indefinitely. Learn Math for Free. Visit www.ganitgurooz.com
10. 10. How to represent a set in Set-Builder Form? Class - XI MB1101: Sets Important Definitions Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com
11. 11. How to represent a set in Set-Builder Form? Class - XI MB1101: Sets Important Definitions Topic: Representations of Sets In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set { , , , , }, all the elements possessa e i o u a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by , we write = { : is a vowel in English alphabet}. V V x x Learn Math for Free. Visit www.ganitgurooz.com
12. 12. What is Empty set? Class - XI MB1101: Sets Important Definitions Topic: Empty Set Learn Math for Free. Visit www.ganitgurooz.com
13. 13. What is Empty set? Class - XI MB1101: Sets Important Definitions Topic: Empty Set We define empty set also called null or void set, as a set which does not contain any element. We denote empty set by the symbol . Learn Math for Free. Visit www.ganitgurooz.com
14. 14. What are Finite and Infinite sets? Class - XI MB1101: Sets Important Definitions Topic: Finite and Infinite Sets Learn Math for Free. Visit www.ganitgurooz.com
15. 15. What are Finite and Infinite sets? Class - XI MB1101: Sets Important Definitions Topic: Finite and Infinite Sets A set which is either empty or contains a definite number of element, is called a finite set. A set which is not finite is called is an infinite set e.g. the set of all natural numbers is an infinite set and 1,2,3 is a finite set .A Learn Math for Free. Visit www.ganitgurooz.com
16. 16. What are equal sets? Class - XI MB1101: Sets Important Definitions Topic: Equal Sets Learn Math for Free. Visit www.ganitgurooz.com
17. 17. What are equal sets? Class - XI MB1101: Sets Important Definitions Topic: Equal Sets Given two sets and , if every element of is also an element of and if every element of is also an element of , then the sets and are said to be equal. In this case we write = . If two A B A B B A A B A B sets are equal then they have exactly the same elements. If two sets are not equal then they are called as unequal sets and we write A B Learn Math for Free. Visit www.ganitgurooz.com
18. 18. What are Subsets? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
19. 19. What are Subsets? Class - XI MB1101: Sets Important Definitions Topic: Subsets A set is said to be a subset of a set if every element of is also an element of . In other words, if whenever , then . We use the symbol " " which means implies. Using this symbol, we A B A B A B a A a B can write the definition of as follows: if . We read the above statement as " is a subset of if is an element of implies that is also an element of ". If is not a subset of subset A B a A a B A B A A B A , we write . B A B Learn Math for Free. Visit www.ganitgurooz.com
20. 20. What are Subsets? Class - XI MB1101: Sets Important Definitions Topic: Subsets We use the symbol " " for two way implications, and read it as if and only also written as "iff"). So and It follows from the above definition that every set , i.e., A B A B B A A is a subset of itself . Since the empty set has no elements, we agree to say that is a subset of every set. We now consider some examples : Some of the obvious relations among the subsets of real numbers are A A N Z Q R Learn Math for Free. Visit www.ganitgurooz.com
21. 21. What is Super Set? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
22. 22. What is Super Set? Class - XI MB1101: Sets Important Definitions Topic: Subsets Let and be two sets. If and , then is called a proper subset of and is called a superset of A. For example, = {1, 2, 3} is a proper subset of = {1, 2, 3, 4}. A B A B A B A B B A B Learn Math for Free. Visit www.ganitgurooz.com
23. 23. What is Singleton Set? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
24. 24. What is Singleton Set? Class - XI MB1101: Sets Important Definitions Topic: Subsets If a set has only one element, we call it a . Thus,{ a } is a singleton set. A singleton set Learn Math for Free. Visit www.ganitgurooz.com
25. 25. Explain Intervals as a subset of R? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
26. 26. Describe Intervals as a subset of R? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com Let , R and . Then the set of real numbers : is called an open interval and is denoted by ( , ). All the points between and belong to the open interval ( , ) but , themselves do n a b a b y a y b a b a b a b a b ot belong to this interval. The interval which contains the end points also is called and is denoted by [ , ]. Thus, , : . We can also have intervals closed at one end and open closed interval a b a b x a x b at the other, i.e., , : is an from to , including but excluding . , : is an open interval from to including but excluding . a b x a x b open interval a b a b a b x a x b a b b a
27. 27. Describe Intervals as a subset of R? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com These notations provide an alternative way of designating the subsets of set of real numbers. For example, if 3, 5 and 7, 9 , then . The set 0, defines the set of non-negative real numbers, A B A B while set ,0 defines the set of negative real numbers. The set , describes the set of real numbers as a line extending from to . The number ( ) is called the length of any of the intervals b a ( , ), [ , ], [ , ) or ( , ]. a b a b a b a b
28. 28. Describe Intervals as a subset of R? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com On real number line, various types of intervals described above as subsets of , are shown below: Graphically, an open interval 3,1 is shown as: R The interval which contains the end points also is called closed interval and is denoted by , Thus, , : . Graphically, a closed interval 3,1 is shown as: a b a b x a x b
29. 29. Describe Intervals as a subset of R? Class - XI MB1101: Sets Important Definitions Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com We can also have closed at one end and open at the other, i.e., , : is an half-open interval from to , including but excluding . , : is an half-open interval f half - open intervals a b x a x b a b a b a b x a x b rom to including but excluding . So, the half open interval 2,4 is shown graphically as: a b b a
30. 30. What is Power Set? Class - XI MB1101: Sets Important Definitions Topic: Power Set Learn Math for Free. Visit www.ganitgurooz.com
31. 31. What is Power Set? Class - XI MB1101: Sets Important Definitions Topic: Power Set The collection of all subsets of a set is called the of . It is denoted by ( ). In ( ), every element is a set. Thus, as in above, if A = { 1, 2 }, then ( ) = { , { 1 }, { 2 }, { 1,2 }} A power set A P A P A P A 2 . Also, note that the number of elements in ( ) = [ ( )] = 4 = 2 . In general, if is a set with ( ) = , then it can be shown that [ ( )] = 2 .m P A n P A A n A m n P A Learn Math for Free. Visit www.ganitgurooz.com
32. 32. What is Universal Set? Class - XI MB1101: Sets Important Definitions Topic: Universal Set Learn Math for Free. Visit www.ganitgurooz.com
33. 33. What is Universal Set? Class - XI MB1101: Sets Important Definitions Topic: Universal Set A universal set is the collection of all objects in a particular context. All other sets in that framework constitute subsets of the universal set, which is denoted as an uppercase italic letter . Th U e objects themselves are known as elements or members of .U Learn Math for Free. Visit www.ganitgurooz.com
34. 34. What are Venn Diagrams? Class - XI MB1101: Sets Important Definitions Topic: Venn Diagrams Learn Math for Free. Visit www.ganitgurooz.com
35. 35. What are Venn Diagrams? Class - XI MB1101: Sets Important Definitions Topic: Venn Diagrams Learn Math for Free. Visit www.ganitgurooz.com Most of the relationships between sets can be represented by means of diagrams which are known as . In Venn diagrams, the elements of the sets are written in their respective circles. The Venn diagrams Venn diagram with three overlapping circle makes 7 different regions as shown below:
36. 36. What are Venn Diagrams? Class - XI MB1101: Sets Important Definitions Topic: Venn Diagrams Learn Math for Free. Visit www.ganitgurooz.com Let us try to understand what these regions indicate: Region 1 contains elements that are in but not in or . Thus 1, if , and . Region 2 contains elements that are in but not in A B C x x A x B x C B A or . Thus 2, if , and . Region 3 contains elements that are and but not in . Thus 3, if , and . Region 4 contains elements that are in but not in or . Thus 4, i C x x B x A x C A B C x x A x B x C C A B x f , and . Region 5 contains elements that are and but not in . Thus 5, if , and . Region 6 contains elements that are and but not in . Thus 6, if , and . R x C x A x B A C B x x A x C x B B C A x x B x C x A egion 7 contains elements that are in , and . Thus 7, if , and . A B C x x A x B x C
37. 37. What is Union of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
38. 38. What is Union of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com Let and be any two sets. The union of and is the set which consists of all the elements of and all the elements of , the common elements being taken only once. The symbol ' ' is used to denot A B A B A B e the . Symbolically, we write and usually read as ' union '. Thus, we can define the union of two sets and as the set C which consists of all those elements which are either in A or in union A B A B A B B (including those which are in both). In symbols, we write : or .A B x x A x B
39. 39. What is Union of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com The union of two sets can be represented by a Venn diagram as shown in the figure: The shaded portion in Figure represents .A B
40. 40. What is Intersection of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
41. 41. What is Intersection of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com The intersection of sets and is the set of all elements which are common to both and . The symbol ' ' is used to denote the . The intersection of two sets and is the set of all t A B A B intersection A B hose elements which belong to both and . Symbolically, we write : and . Thus, we can define the intersection of two sets and as the set of all those elements which belong both A and B. A B A B x x A x B A B
42. 42. What is Intersection of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com The shaded portion in Figure indicates the intersection of A and B.
43. 43. What is Difference of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
44. 44. What is Difference of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com The difference of the sets and in this order is the set of elements which belong to but not to . Symbolically, we write and read as" minus ". Using the set builder notation, we can rewrit A B A B A B A B e the definition of difference as : and . The difference of two sets and can be represented by Venn diagram as shown in the figure. A B x x A x B A B The shaded portion represents the difference of the two sets and .A B
45. 45. What is Difference of Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com The sets , and are mutually disjoint sets, i.e., the intersection of any of these two sets is the null set as shown in the figure A B A B B A
46. 46. What is Complement of a Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
47. 47. What is Complement of a Sets? Class - XI MB1101: Sets Important Definitions Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com Let be the universal set and is a subset of . Then the complement of is the set of all elements of which are not the elements of A. Symbolically, we write ' to denote the complement of wit U A U A U A A h respect to . Thus, : and . Obviously . So the complement of a set can also be seen as difference between a universal set and the set A. U A x x U x A A U A A
48. 48. Fact # 1 Two sets A and B are said to be equal if they have exactly the same elements. Class - XI MB1101: Sets Important Facts Topic: Equal Sets Learn Math for Free. Visit www.ganitgurooz.com
49. 49. Fact # 2 A set A is said to be subset of a set B, if every element of A is also an element of B. Class - XI MB1101: Sets Important Facts Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
50. 50. Fact # 3 Intervals are subsets of R. Class - XI MB1101: Sets Important Facts Topic: Subsets Learn Math for Free. Visit www.ganitgurooz.com
51. 51. Fact # 4 A power set of a set A is collection of all subsets of A. It is denoted by P(A). Class - XI MB1101: Sets Important Facts Topic: Power Sets Learn Math for Free. Visit www.ganitgurooz.com
52. 52. Fact # 5 The union of two sets A and B is the set of all those elements which are either in A or in B. Class - XI MB1101: Sets Important Facts Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
53. 53. Fact # 6 The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B. Class - XI MB1101: Sets Important Facts Topic: Operations on Sets Learn Math for Free. Visit www.ganitgurooz.com
54. 54. Fact # 7 The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A. Class - XI MB1101: Sets Important Facts Topic: Complement of a Set Learn Math for Free. Visit www.ganitgurooz.com
55. 55. Class - XI MB1101: Sets Important Formulae Topic: Complement of a Set 1. Complement law: ( ) ( ) i A A U ii A A Learn Math for Free. Visit www.ganitgurooz.com
56. 56. Class - XI MB1101: Sets Important Formulae Topic: Complement of a Set 2. De Morgan's law: ( ) ( ) i A B A B ii A B A B Learn Math for Free. Visit www.ganitgurooz.com
57. 57. Class - XI MB1101: Sets Important Formulae Topic: Complement of a Set 3. Law of double complementation: A A Learn Math for Free. Visit www.ganitgurooz.com
58. 58. Class - XI MB1101: Sets Important Formulae Topic: Complement of a Set 4. Laws of empty set and universal set and .U U Learn Math for Free. Visit www.ganitgurooz.com
59. 59. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 5. Commutative law ,A B B A A B B A Learn Math for Free. Visit www.ganitgurooz.com
60. 60. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 6. Law of identity element , ,A A A U A A Learn Math for Free. Visit www.ganitgurooz.com
61. 61. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 7. Idempotent law ,A A A A A A Learn Math for Free. Visit www.ganitgurooz.com
62. 62. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 8. U A U Learn Math for Free. Visit www.ganitgurooz.com
63. 63. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 9. Associative Laws: ,A B C A B C A B C A B Learn Math for Free. Visit www.ganitgurooz.com
64. 64. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 10. Distributive Laws: ,A B C A B A C A B C A B A C Learn Math for Free. Visit www.ganitgurooz.com
65. 65. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 11. Let and be finite sets. If thenA B A B n A B n A n B Learn Math for Free. Visit www.ganitgurooz.com
66. 66. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 12. In general, if and are finite sets, thenA B .n A B n A n B n A B Learn Math for Free. Visit www.ganitgurooz.com
67. 67. Class - XI MB1101: Sets Important Formulae Topic: Operations on Sets 13. For any three finite sets , ,A B C . n A B C n A n B n C n A B n B C n A C n A B C Learn Math for Free. Visit www.ganitgurooz.com
68. 68. How to change the representation of the set from Set- Builder to Roster Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com MB1101: Sets
69. 69. How to change the representation of the set from Set- Builder to Roster Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com :Understand the description of the elements of the set.Step1 MB1101: Sets
70. 70. How to change the representation of the set from Set- Builder to Roster Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com :Understand the description of the elements of the set.Step1 MB1101: Sets :Identify each member of the set one at a time.Step2
71. 71. How to change the representation of the set from Set- Builder to Roster Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com :Understand the description of the elements of the set.Step1 MB1101: Sets :Identify each member of the set one at a time.Step2 : If you can see a pattern after finding a few members, use an ellipsis to write the set in Roster form so that everyone can see the pattern and guess ALL elements. Step3
72. 72. How to change the representation of the set from Set- Builder to Roster Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com :Understand the description of the elements of the set.Step1 MB1101: Sets :Identify each member of the set one at a time.Step2 : If you can see a pattern after finding a few members, use an ellipsis to write the set in Roster form so that everyone can see the pattern and guess ALL elements. Step3 : If there are just a few elements (usually less than 10) then you need to mention all elements in the Roster form. Step4
73. 73. How to change the representation of the set from Roster to Set-Builder Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com MB1101: Sets
74. 74. How to change the representation of the set from Roster to Set-Builder Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com MB1101: Sets :Find a pattern among all members of the set.Step1
75. 75. How to change the representation of the set from Roster to Set-Builder Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com MB1101: Sets :Find a pattern among all members of the set.Step1 :Find out what are the common characteristics of the members.Step2
76. 76. How to change the representation of the set from Roster to Set-Builder Form? Class – XI,CBSE Important Procedures Topic: Representations of Sets Learn Math for Free. Visit www.ganitgurooz.com MB1101: Sets :Find a pattern among all members of the set.Step1 :Find out what are the common characteristics of the members.Step2 : Use a group of statements to describe these common characteristics in set-builder form. This conversion is not always easy because the description of the elements of the set may not be unique. Step3