1. Set Theory

2. BASIC CONCEPTS OF SETS Content Define a set Represent a set Define the number of elements of a set Practice Problems 1 Compare sets Recognize different types of set Practice Problems 2 Define the subsets of a set Review Exercises 1 OPERATIONS ON SETS Determine the union of two or more sets Practice Problems 3 Determine the intersection of two or more sets Practice Problems 4 Determine the complement of a set Determine the difference of two sets Practice Problems 5 Review Exercises 2 Lesson is over

3. Defining a Set Definition A collection of well-defined objects is called a set . ‘ the set of former Nobel Prize winners’ is a well-defined set ‘ the set of tall students in our university’ is not a well-defined set For example, It’s interesting  Note

4. Note Sets must be well-defined. A set is well-defined if we can tell whether a particular object is an element of that set.

5. Definition Each object in a set is called an element or a member of the set. Defining a Set A, B, C, … a, b, c, … Sets notation: Elements Notation:  Note For example, A = {a, b, c, d} b  A ‘ b is an element of set A’ or ‘b is in A’ f  A ‘ f is not an element of set A’ or ‘f is not in A’ Example 1 Check

6. The order of elements in a set is not important. Note Each element of the set is written only once.

7. George Cantor (1845-1918) George Cantor was born March 3, 1845 in St. Petersburg, Russia and died January 6, 1918 in Halle, Germany. Cantor was the first mathematician who defined the basic ideas of set theory. Using ingenious methods, he proved remarkable things about infinite sets.

8. Defining a Set Solution 1 h  B h. 1  B g. 3  B f. 10  B e. 0  B d.   B c.   B b. a  B a.

10. Representing Sets The List Method For instance, A = {  ,  ,  ,  } Example 2 C = {I, You, He, She, We, They} Check

12. Representing Sets The Defining-Property Method For example, B={x| x is a season of the year} When we use the defining-property method we can use the symbol ‘:’ instead of ‘|’. Example 3 Check B is the set of all x such that x is a season of the year  Note

14. Representing Sets The Venn Diagram Method . c As an example, . b . a A Each element of the set is represented by a point inside the closed shape. Example 4 Check  Note

15. John Venn (1834-1923) John Venn was born August 4, 1834 in Hull, Yorkshire, England and died April 4, 1923 in Cambridge, England. Venn diagrams (a diagrammatic way of representing sets & their relationships) were introduced in 1880 by John Venn.

16. Representing Sets Solution 4 A . 9 . 7 . 5 . 3 . 1 a. b. . July . June . January . 5 C . 0 . 10 . 20 . 15 c.

17. Number of Elements of a Set Definition Denotation: The number of elements in a set A is called the cardinal number of the set A. n(A) Example 5 Check

19. Practice Problems 1 f  A A = {  ,  ,  ,  } . c . b . a A n(A) B={x| x is a season of the year}

21. Comparing Sets Definition Two sets are equivalent if they have the same number of elements. If A & B are equivalent A  B ‘ set A is equivalent to set B’ For example , A = {a, b, c, d} & B = {1, 2, 3, 4} n(A) = n(B) = 4 so A  B Equivalent Sets

22. Comparing Sets Definition Two sets are equal if they have exactly the same elements. If A & B are equal A = B ‘ set A is equal to set B’ Example 6 Check Equal Sets

23. A & B contain exactly the same elements, therefore they are equal. Solution 6 Comparing Sets Therefore, we can write A = B.

24. Note If A is equal to B, then A is also equivalent to B. A = B  A  B If A is equivalent to B, then A might not be equal to B. However, A  B  A = B

26. Types of Set Empty (Null) Set Definition The set that contains no element is called the empty set or the null set . Denotation:  or { } As an example, A = {x| x is a month containing 32 days} So A =  & n(  ) = 0

27. Types of Set The Universal Set Definition The Universal set is the set of all elements under consideration in a given discussion. Denotation: U For instance, U = {a, b, c, d, e, f, g, h, I, j} A = {a, b, c} B = {c, d, e} C = {f, g, h}

28. Types of Set Finite & Infinite Sets Definition If the number of elements in a set is a whole number , the set is a finite set . If a set is not finite then it is an infinite set . Example 7 Check As an example, ‘ the set of days of the week’ is a finite set ‘ the set of whole numbers’ is an infinite set

30. Practice Problems 2 A = B  A  B A = B n(  ) = 0 U = {a, b, c, d, e, f, g, h, I, j} Z = {… -3, -2, -1, 0, 1, 2, 3, …}

34. REVIEW EXERCISES 1 S = {  ,  ,  ,  } A = B  A  B . c . b . a D n(  ) = 0 f  A

35. Union of Sets Definition The union of sets A & B is the set of all elements which belong to A or to B (or to both). We write A  B & say ‘ A union B’ Union of sets A and B A  B = {a, b, c, d, e, f, g, h}. If A = {a, b, c, d, e} & B = {a, c, f, g, h}, then Example 9 Check  Note

36. Note If a  A  B then a  A or a  B. Each element must be written only once in the union of sets.

40. Practice Problems 3 A  B = B  A A  B n(  ) = 0 U = {a, b, c, d, e, f, g, h, I, j} Z = {… -3, -2, -1, 0, 1, 2, 3, …}

41. Intersection of Sets Definition The intersection of two sets A & B is the set of all elements that are in both A & B .  Note We write A  B & say ‘ A intersection B’ Example 11 Check

42. Note If a  A  B then a  A & a  B.

48. Practice Problems 4 A  (B  C) A  B = B  A A  B n(  ) = 0

49. Complement of a Set Definition The set of elements of U which are not elements of set A is called the complement of A. A c or A’ ‘ the complement of set A’

52. Difference of Two Sets Definition The set of elements that are in A but not in B is called the difference of sets A and B .  Note A-B or A ‘ A difference B ’ Example 15 Check

53. Note

57. Practice Problems 5 A  B = B  A A  B n(  ) = 0 A  (B  C)

58. REVIEW EXERCISES 2 n(  ) = 0 A  (B  C) A  A’ = U A – A = 