Chapter 2

1. Chapter 2 The Basic Concepts of Set Theory © 2008 Pearson Addison-Wesley. All rights reserved

2. Chapter 2: The Basic Concepts of Set Theory 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Surveys and Cardinal Numbers 2.5 Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved

3. Chapter 1 Section 2-1 Symbols and Terminology © 2008 Pearson Addison-Wesley. All rights reserved

4. Symbols and Terminology • Designating Sets • Sets of Numbers and Cardinality • Finite and Infinite Sets • Equality of Sets © 2008 Pearson Addison-Wesley. All rights reserved

5. Designating Sets A set is a collection of objects. The objects belonging to the set are called the elements, or members of the set. • Sets are designated using: • 1) word description, • the listing method, and • set-builder notation. © 2008 Pearson Addison-Wesley. All rights reserved

6. Designating Sets Word description The set of even counting numbers less than 10 The listing method {2, 4, 6, 8} Set-builder notation {x|x is an even counting number less than 10} © 2008 Pearson Addison-Wesley. All rights reserved

7. Designating Sets Sets are commonly given names (capital letters). A = {1, 2, 3, 4} The set containing no elements is called the empty set (null set) and denoted by { } or To show 2 is an element of set A use the symbol © 2008 Pearson Addison-Wesley. All rights reserved

8. Example: Listing Elements of Sets Give a complete listing of all of the elements of the set {x|x is a natural number between 3 and 8} Solution {4, 5, 6, 7} © 2008 Pearson Addison-Wesley. All rights reserved

9. Sets of Numbers Natural (counting) {1, 2, 3, 4, …} Whole numbers {0, 1, 2, 3, 4, …} Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational numbers May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333… Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat. Real numbers {x | x can be expressed as a decimal} © 2008 Pearson Addison-Wesley. All rights reserved

10. Cardinality The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A), read “n of A,” represents the cardinal number of set A. © 2008 Pearson Addison-Wesley. All rights reserved

11. Example: Cardinality Find the cardinal number of each set. a) K = {a, l, g, e, b, r} b) M = {2} c) © 2008 Pearson Addison-Wesley. All rights reserved

12. Finite and Infinite Sets If the cardinal number of a set is a particular whole number, we call that set a finite set. Whenever a set is so large that its cardinal number is not found among the whole numbers, we call that set an infinite set. © 2008 Pearson Addison-Wesley. All rights reserved

13. Example: Infinite Set The odd counting numbers are an infinite set. Word description The set of all odd counting numbers Listing method {1, 3, 5, 7, 9, …} Set-builder notation {x|x is an odd counting number} © 2008 Pearson Addison-Wesley. All rights reserved

14. Equality of Sets Set A is equal to set B provided the following two conditions are met: 1. Every element of A is an element of B, AND 2. Every element of B is an element of A. © 2008 Pearson Addison-Wesley. All rights reserved

15. Example: Equality of Sets State whether the sets in each pair are equal. a) {a, b, c, d} and {a, c, d, b} b) {2, 4, 6} and {x|x is an even number} © 2008 Pearson Addison-Wesley. All rights reserved

16. Section 2.1: Symbols and Terminology • Which of the following is an example of set- • builder notation? • a) The counting numbers less than 5 • b) {1, 2, 3, 4} • c) {x | x is a counting number less than 5} © 2008 Pearson Addison-Wesley. All rights reserved

17. Section 2.1: Symbols and Terminology 2. Are sets {6, 7, 8} and {7, 8, 6} equal? a) Yes b) No © 2008 Pearson Addison-Wesley. All rights reserved

18. Chapter 1 Section 2-2 Venn Diagrams and Subsets © 2008 Pearson Addison-Wesley. All rights reserved

19. Venn Diagrams and Subsets • Venn Diagrams • Complement of a Set • Subsets of a Set • Proper Subsets • Counting Subsets © 2008 Pearson Addison-Wesley. All rights reserved

20. Venn Diagrams In set theory, the universe of discourse is called the universal set, typically designated with the letter U. Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these diagrams, the universal set is represented by a rectangle and other sets of interest within the universal set are depicted as circular regions. © 2008 Pearson Addison-Wesley. All rights reserved

21. Venn Diagrams The rectangle represents the universal set, U, while the portion bounded by the circle represents set A. A U © 2008 Pearson Addison-Wesley. All rights reserved

22. Complement of a Set The colored region inside U and outside the circle is labeled A'(read “Aprime”). This set, called the complement of A, contains all elements that are contained in U but not in A. A U © 2008 Pearson Addison-Wesley. All rights reserved

23. Complement of a Set For any set A within the universal set U, the complement of A, written A',is the set of all elements of U that are not elements of A. That is © 2008 Pearson Addison-Wesley. All rights reserved

24. Subsets of a Set Set A is a subset of set B if every element of A is also an element of B. In symbols this is written B A U © 2008 Pearson Addison-Wesley. All rights reserved

25. Example: Subsets Fill in the blank with to make a true statement. a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} © 2008 Pearson Addison-Wesley. All rights reserved

26. Set Equality (Alternative Definition) Suppose that A and B are sets. Then A = B if © 2008 Pearson Addison-Wesley. All rights reserved

27. Proper Subset of a Set Set A is a proper subset of set B if In symbols, this is written © 2008 Pearson Addison-Wesley. All rights reserved

28. Example: Proper Subsets Decide whether or both could be placed in each blank to make a true statement. a) {a, b, c} ___ { a ,b, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} © 2008 Pearson Addison-Wesley. All rights reserved

29. Counting Subsets One method of counting subsets involves using a tree diagram. The figure below shows the use of a tree diagram to find the subsets of {a, b}. Yes No {a, b} {a} {b} Yes No Yes No © 2008 Pearson Addison-Wesley. All rights reserved

30. Number of Subsets The number of subsets of a set with n elements is 2n. The number of proper subsets of a set with n elements is 2n – 1. © 2008 Pearson Addison-Wesley. All rights reserved

31. Example: Number of Subsets Find the number of subsets and the number of proper subsets of the set {m, a, t, h, y}. © 2008 Pearson Addison-Wesley. All rights reserved

32. Section 2.2: Venn Diagrams and Subsets 1. Find the complement of {m} if U = {m, n}. a) {m} b) {n} c) U d) © 2008 Pearson Addison-Wesley. All rights reserved

33. Section 2.2: Venn Diagrams and Subsets 2. Find the number of subsets of { $, #, @ }. a) 0 b) 3 c) 6 d) 8 © 2008 Pearson Addison-Wesley. All rights reserved

34. Chapter 1 Section 2-3 Set Operations and Cartesian Products © 2008 Pearson Addison-Wesley. All rights reserved

35. Set Operations and Cartesian Products • Intersection of Sets • Union of Sets • Difference of Sets • Ordered Pairs • Cartesian Product of Sets • Venn Diagrams • De Morgan’s Laws © 2008 Pearson Addison-Wesley. All rights reserved

36. Intersection of Sets The intersection of sets A and B, written is the set of elements common to both A and B, or © 2008 Pearson Addison-Wesley. All rights reserved

37. Example: Intersection of Sets Find each intersection. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

38. Union of Sets The union of sets A and B, written is the set of elements belonging to either of the sets, or © 2008 Pearson Addison-Wesley. All rights reserved

39. Example: Union of Sets Find each union. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

40. Difference of Sets The difference of sets A and B, written A – B, is the set of elements belonging to set A and not to set B, or © 2008 Pearson Addison-Wesley. All rights reserved

41. Example: Difference of Sets Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, h}, B = {c, e, g}, and C = {a, c, d, g, e}. Find each set. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

42. Ordered Pairs In the ordered pair (a, b), a is called the first component and b is called the second component. In general Two ordered pairs are equal provided that their first components are equal and their second components are equal. © 2008 Pearson Addison-Wesley. All rights reserved

43. Cartesian Product of Sets The Cartesian product of sets A and B, written, is © 2008 Pearson Addison-Wesley. All rights reserved

44. Example: Finding Cartesian Products Let A = {a, b}, B = {1, 2, 3} Find each set. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

45. Cardinal Number of a Cartesian Product If n(A) = a and n(B) = b, then © 2008 Pearson Addison-Wesley. All rights reserved

46. Example: Finding Cardinal Numbers of Cartesian Products If n(A) = 12and n(B) = 7, then find © 2008 Pearson Addison-Wesley. All rights reserved

47. Venn Diagrams of Set Operations A B A B U U A A B U U © 2008 Pearson Addison-Wesley. All rights reserved

48. Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set © 2008 Pearson Addison-Wesley. All rights reserved

49. Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set © 2008 Pearson Addison-Wesley. All rights reserved

50. De Morgan’s Laws For any sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved