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Chapter 2

Chapter 2. The Basic Concepts of Set Theory. 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. Section 2-1. Symbols and Terminology.

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Chapter 2

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  1. Chapter 2 The Basic Concepts of Set Theory 2012 Pearson Education, Inc.

  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 2012 Pearson Education, Inc.

  3. Section 2-1 • Symbols and Terminology 2012 Pearson Education, Inc.

  4. Symbols and Terminology • Designating Sets • Sets of Numbers and Cardinality • Finite and Infinite Sets • Equality of Sets 2012 Pearson Education, Inc.

  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, 2) the listing method, and 3) set-builder notation. 2012 Pearson Education, Inc.

  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} 2012 Pearson Education, Inc.

  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 2012 Pearson Education, Inc.

  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} 2012 Pearson Education, Inc.

  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} 2012 Pearson Education, Inc.

  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. 2012 Pearson Education, Inc.

  11. Example: Cardinality Find the cardinal number of each set. a) K = {a, l, g, e, b, r} b) M = {2} c) Solution a) n(K) = 6 b) n(M) = 1 c) 2012 Pearson Education, Inc.

  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. 2012 Pearson Education, Inc.

  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} 2012 Pearson Education, Inc.

  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. 2012 Pearson Education, Inc.

  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} Solution a) Yes, order of elements does not matter b) No, {2, 4, 6} does not represent all the even numbers. 2012 Pearson Education, Inc.

  16. 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 2012 Pearson Education, Inc.

  17. Section 2-2 • Venn Diagrams and Subsets 2012 Pearson Education, Inc.

  18. Venn Diagrams and Subsets • Venn Diagrams • Complement of a Set • Subsets of a Set • Proper Subsets • Counting Subsets 2012 Pearson Education, Inc.

  19. 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. 2012 Pearson Education, Inc.

  20. A U Venn Diagrams The rectangle represents the universal set, U, while the portion bounded by the circle represents set A. 2012 Pearson Education, Inc.

  21. A U 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. 2012 Pearson Education, Inc.

  22. 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 2012 Pearson Education, Inc.

  23. B A U 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 2012 Pearson Education, Inc.

  24. 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} Solution a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} 2012 Pearson Education, Inc.

  25. Set Equality (Alternative Definition) Suppose that A and B are sets. Then A = B if 2012 Pearson Education, Inc.

  26. Proper Subset of a Set Set A is a proper subset of set B if In symbols, this is written 2012 Pearson Education, Inc.

  27. 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} Solution a) both b) 2012 Pearson Education, Inc.

  28. 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 2012 Pearson Education, Inc.

  29. 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. 2012 Pearson Education, Inc.

  30. Example: Number of Subsets Find the number of subsets and the number of proper subsets of the set {m, a, t, h, y}. Solution Since there are 5 elements, the number of subsets is 25 = 32. The number of proper subsets is 32 – 1 = 31. 2012 Pearson Education, Inc.

  31. 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 2012 Pearson Education, Inc.

  32. Section 2-3 • Set Operations and Cartesian Products 2012 Pearson Education, Inc.

  33. 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 2012 Pearson Education, Inc.

  34. Intersection of Sets The intersection of sets A and B, written is the set of elements common to both A and B, or 2012 Pearson Education, Inc.

  35. Example: Intersection of Sets Find each intersection. a) b) Solution a) b) 2012 Pearson Education, Inc.

  36. Union of Sets The union of sets A and B, written is the set of elements belonging to either of the sets, or 2012 Pearson Education, Inc.

  37. Example: Union of Sets Find each union. a) b) Solution a) b) 2012 Pearson Education, Inc.

  38. 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 2012 Pearson Education, Inc.

  39. 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) Solution a) {a, b, h} b) 2012 Pearson Education, Inc.

  40. 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. 2012 Pearson Education, Inc.

  41. Cartesian Product of Sets The Cartesian product of sets A and B, written, is 2012 Pearson Education, Inc.

  42. Example: Finding Cartesian Products Let A = {a, b}, B = {1, 2, 3} Find each set. a) b) Solution a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} 2012 Pearson Education, Inc.

  43. Cardinal Number of a Cartesian Product If n(A) = a and n(B) = b, then 2012 Pearson Education, Inc.

  44. Example: Finding Cardinal Numbers of Cartesian Products If n(A) = 12and n(B) = 7, then find Solution 2012 Pearson Education, Inc.

  45. A B A B U U A B U Venn Diagrams of Set Operations A U 2012 Pearson Education, Inc.

  46. A B U Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set Solution 2012 Pearson Education, Inc.

  47. B A C U Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set Solution 2012 Pearson Education, Inc.

  48. De Morgan’s Laws For any sets A and B, 2012 Pearson Education, Inc.

  49. 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 2012 Pearson Education, Inc.

  50. Section 2-4 • Surveys and Cardinal Numbers 2012 Pearson Education, Inc.

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