Chapter 2 The Basic Concepts of Set Theory
Chapter 2 The Basic Concepts of Set Theory © 2008 Pearson Addison-Wesley. All rights reserved
Chapter 2: The Basic Concepts of Set Theory 2. 1 2. 2 2. 3 2. 4 2. 5 Symbols and Terminology Venn Diagrams and Subsets Set Operations and Cartesian Products Surveys and Cardinal Numbers Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved 2
Chapter 1 Section 2 -3 Set Operations and Cartesian Products © 2008 Pearson Addison-Wesley. All rights reserved
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 4
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 5
Example: Intersection of Sets Find each intersection. a) b) Solution a) b) © 2008 Pearson Addison-Wesley. All rights reserved 6
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 7
Example: Union of Sets Find each union. a) b) Solution a) b) © 2008 Pearson Addison-Wesley. All rights reserved 8
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 9
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) © 2008 Pearson Addison-Wesley. All rights reserved 10
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 11
Cartesian Product of Sets The Cartesian product of sets A and B, written, is © 2008 Pearson Addison-Wesley. All rights reserved 12
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)} © 2008 Pearson Addison-Wesley. All rights reserved 13
Cardinal Number of a Cartesian Product If n(A) = a and n(B) = b, then © 2008 Pearson Addison-Wesley. All rights reserved 14
Example: Finding Cardinal Numbers of Cartesian Products If n(A) = 12 and n(B) = 7, then find Solution © 2008 Pearson Addison-Wesley. All rights reserved 15
Venn Diagrams of Set Operations A B B A B U U A U © 2008 Pearson Addison-Wesley. All rights reserved 16
Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set Solution A B U © 2008 Pearson Addison-Wesley. All rights reserved 17
Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set Solution B A U C © 2008 Pearson Addison-Wesley. All rights reserved 18
De Morgan’s Laws For any sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved 19
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