General Mathematis with the Topic of SETs Story

General Mathematics Concepts

1. GENERAL MATHEMATICS SET THEORY STATE COLLEGE OF EDUCATION MIRPUR

2. INTRODUCTION TO SETS A set is a well-defined collection of distinct objects. Well-defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally, sets are denoted by capital letters A, B, C, X, Y, Z etc. Sets are fundamental objects in mathematics used to define relationships between elements. Sets can be finite or infinite, depending on the number of elements they contain.

3. ELEMENTS OF A SET The objects or elements in a set can be anything, such as numbers, letters, or even other sets. Elements of a set are denoted by listing them within curly braces, separated by commas. Each element in a set is unique, and the order of elements does not matter.

4. Symbols Meaning { } Symbol of set U Universal set n(X) Cardinal number of set X b ∈ A 'b' is an element of set A a ∉ B 'a' is not an element of set B ∅ Null or  empty set A U B Set A  union  set B A ∩ B Set A  intersection  set B A ⊆ B Set A is a  subset  of set B B ⊇ A Set B is the  superset of set A SETS SYMBOLS

5. TYPES OF SET We have several types of sets in Math's. They are empty set, finite and infinite sets, proper set, equal sets, etc. Let us go through the classification of sets here. Empty Set: A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø . A set of apples in the basket of grapes is an example of an empty set because in a grapes basket there are no apples present . Singleton Set: A set which contains a single element is called a singleton set. Example: There is only one apple in a basket of grapes.

6. Finite set: A set which consists of a definite number of elements is called a finite set. Example: A set of natural numbers up to 10. A = {1,2,3,4,5,6,7,8,9,10 } Infinite set: A set which is not finite is called an infinite set. Example: A set of all natural numbers. A = {1,2,3,4,5,6,7,8,9……} TYPES OF SET

7. TYPES OF SET Equivalent Set: If the number of elements is the same for two different sets, then they are called equivalent sets. The order of sets does not matter here. It is represented as:  n(A) = n(B ) where A and B are two different sets with the same number of elements . Example: If A = {1,2,3,4} and B = {Red, Blue, Green, Black} In set A, there are four elements and in set B also there are four elements. Therefore, set A and set B are equivalent.

8. Equal sets: The two sets A and B are said to be equal if they have exactly the same elements, the order of elements do not matter. Example: A = {1,2,3,4} and B = {4,3,2,1} A = B Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element. Example: Set A = {1,2,3,4} and set B = {5,6,7,8} are disjoint sets, because there is no common element between them. TYPES OF SET

9. TYPES OF SET Subsets: A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part of another set. Example: A = {1,2,3} Then {1,2} ⊆ A . Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}. Note : The set is also a subset of itself. If A is not a subset of B, then it is denoted as A⊄B.

10. TYPES OF SET Proper Subset: If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B. Example: If A = {2,5,7} is  a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7} But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also. Superset: Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B. For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.

11. TYPES OF SET Universal Set: A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values.  Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be: U = {1,2,3,4,5}

12. OPERATIONS ON SETS In set theory, the operations of the sets are carried when two or more sets combine to form a single set under some of the given conditions. The basic operations on sets are : Union of sets Intersection of sets A complement of a set Set difference Cartesian product of sets. Basically, we work more on   union and intersection of sets  operations, using Venn diagrams.

13. Union of Sets If set A and set B are two sets, then A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B . Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is: A ∪ B = {1,2,3,4,5,6} OPERATIONS ON SETS

14. Intersection of Sets If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B . Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is: A ∩ B = { } or Ø Since A and B do not have any elements in common, so their intersection will give null set. OPERATIONS ON SETS

15. OPERATIONS ON SETS Complement of Sets The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’ Properties of Complement sets P ∪ P′ = U P ∩ P′ = Φ Law of double complement : (P′ )′ = P Laws of empty/null set(Φ) and universal set(U),  Φ′ = U and U′ = Φ.

16. OPERATIONS ON SETS Cartesian Product of S ets If set A and set B are two sets then the Cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B. We can represent it in set-builder form, such as: A × B = {(a, b) : a ∈ A and b ∈ B } Example: set A = {1,2,3} and set B = {Bat, Ball}, then; A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

17. Difference of Sets If set A and set B are two sets, then set A difference set B is a set which has elements of A but no elements of B. It is denoted as A – B . Example: A = {1,2,3} and B = {2,3,4} A – B = {1} OPERATIONS ON SETS

18. REPRESENTATION OF SET The sets are represented in curly braces, {}. For example, {2,3,4} or { a,b,c } or {Bat, Ball, Wickets}. Statement or Descriptive form Roster or Tabular Form Set Builder Form

19. REPRESENTATION OF SET Statement or Descriptive Form: In statement form, the well-defined descriptions of a member of a set are written and enclosed in the curly brackets . For example , the set of even numbers less than 15. In statement form, it can be written as {even numbers less than 15}.

20. Roster or Tabular Form: In Roster form, all the elements of a set are listed . For example, the set of natural numbers less than 5. Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,………. Natural Number less than 5 = 1, 2, 3, 4 Therefore, the set is N = { 1, 2, 3, 4 } REPRESENTATION OF SET

21. REPRESENTATION OF SET Set Builder Form The general form is, A = { x : property } Example: Write the following sets in set builder form: A={2, 4, 6, 8} Solution: 2 = 2 x 1 4 = 2 x 2 6 = 2 x 3 8 = 2 x 4 So, the set builder form is A = {x: x=2n, n ∈ N and 1  ≤ n ≤ 4} Also, Venn Diagrams are the simple and best way for visualized representation of sets.

22. Order of Set: The order of a set defines the number of elements a set is having. It describes the size of a set. The order of set is also known as the  cardinality .  The size of set whether it is is a finite set or an infinite set, said to be set of finite order or infinite order, respectively . The cardinality of the empty set is zero. ORDER OF SET

23. Power Set A power set is set of all subsets, empty set and the original set itself . For example, power set of A = {1, 2} is P(A ) = {{}, {1}, {2}, {1, 2}}. POWER SET

24. PROPERTIES OF SET OPERATIONS

25. :  the set of all natural numbers = {1, 2, 3, 4, .....} :   the set of all whole numbers={0,1 , 2, 3, 4, .....} :   the set of all integers = {0, , , , ,…..} :  the set of all positive integers = {1, 2, 3, 4, .....} :   the set of all negative integers = {.....,-4,- 3, - 2, -1}   SOME IMPORTANT SETS

26. : the set of all even numbers = {0, , , , ,……} the set of all positive even numbers = {2,4,6,8 ,..……} the set of all odd numbers = { , , ,..……} the set of all positive odd numbers = {1,3,5,7,..……} the set of all negative odd numbers = {-1,-3,-5,-7 , ..……}   SOME IMPORTANT SETS

27. SOME IMPORTANT SETS set of all rational numbers : set of all irrational numbers R set of all real numbers =  

28. APPLICATIONS OF SETS Sets are used in various branches of mathematics, including algebra, calculus, and probability. Set theory is foundational in defining mathematical structures and relationships. Sets are also used in computer science, databases, and logic for data organization and analysis.

29. CONCLUSION Sets are essential mathematical structures that help define relationships between elements. Understanding sets and their properties is crucial in various fields of mathematics and beyond. Explore further applications and challenges in set theory to deepen your understanding of this fundamental concept.

30. MCQs

31. A set Q is called a set of Whole numbers Natural numbers Irrational numbers Rational numbers   MCQs

32. The different number of ways to describe a set are 1 2 3 4 MCQs

33. A set with no element is called Subset Empty set Singleton set Super set MCQs

34. The set is Infinite set Subset Null set Finite set   MCQs

35. The set having only one element is called: Null set Power set Singleton set Subset MCQs

36. Power set of an empty set is: , }   MCQs

37. The number of elements in power set is: 4 6 8 9   MCQs

38. If A then A is equal to: A B None of these   MCQs

39. If A then A is equal to: A B None of these   MCQs

40. If A then A is equal to: A B None of these   MCQs

41. is equal to : )   MCQs

42. ) is equal to : ) )   MCQs

43. If A and B are disjoint sets, then is equal to: A B A   MCQs

44. If number of elements in set A is 3 and in set B is 4, then number of elements in A×B is: 3 4 12 7 MCQs

45. If number of elements in se A is 3 and in set B is 2, then number of binary relations in A×B is:   MCQs

46. If , then set A and B are Sub set Over lapping set Disjoint set Power set   MCQs

47. The complement of U is: U Impossible union   MCQs

48. The complement of is: U Impossible union   MCQs

49. = …… U A   MCQs

50. = …… U A   MCQs

51. = …… U A   MCQs

52. The point (−5 , −7) lies in quadrant . I II III IV MCQs

53. y co-ordinate of every point on x–axis is: + ve − ve Zero 1 MCQs

54. x co-ordinate of every point on x–axis is: +ve − ve Zero 1 MCQs

55. The domain of {(a,b), (b,c), ( c,d )} is: { a,b,c } { b,c,d } {a,b } {a , b,c,d ,} MCQs

56. Venn diagram was first used by : John Venn Newton Arthur Clayey John Napie MCQs

57. A subset of A x A is called in A . Set Relation Function Into function MCQs

58. The relation {(a,b),(b,c),(a,d)} is: A function Not a function Range Domain MCQs

59. By definition, which of the following is a set? { a,b,c,a} {1,2,3,2} {l,m,n,o } {0,1,2,3,1} MCQs

60. Which of the following is true? W N Z W N P P W   MCQs

61. Which of the following is true? P N Z W P N W Z P W N Z P Z N W   MCQs

62. Which of the following is true ? N and W Z P and O W O and E W P and E N   MCQs

63. N W =......... {0} N W   MCQs

64. N W =......... {0} N W   MCQs

65. W N =......... ( {0} N W   MCQs

66. Two sets having no common element are called …… sets Subset Overlapping Disjoint Super MCQs

67. If two sets have some elements common but not all are called….sets. Sub Overlapping Disjoint Super MCQs

68. If set A has all its elements common with set B then set A is called….set. Sub Overlapping Disjoint Super MCQs

69. If A is subset of U, then = ……   MCQs

70. THANK YOU ANY QUESTION