1. Chapter 2 Section 3 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 2 Chapter 2 Sets

3. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 3 WHAT YOU WILL LEARN Methods to indicate sets, equal sets, and equivalent sets Subsets and proper subsets Venn diagrams Set operations such as complement, intersection, union, difference and Cartesian product Equality of sets Application of sets Infinite sets

4. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 4 Section 3 Venn Diagrams and Set Operations

5. Chapter 2 Section 3 - Slide 5 Copyright © 2009 Pearson Education, Inc. Venn Diagrams A Venn diagram is a technique used for picturing set relationships. A rectangle usually represents the universal set, U.  The items inside the rectangle may be divided into subsets of U and are represented by circles.

6. Chapter 2 Section 3 - Slide 6 Copyright © 2009 Pearson Education, Inc. Disjoint Sets Two sets which have no elements in common are said to be disjoint. The intersection of disjoint sets is the empty set. Disjoint sets A and B are drawn in this figure. There are no elements in common since there is no overlap- ping area between the two circles.

7. Chapter 2 Section 3 - Slide 7 Copyright © 2009 Pearson Education, Inc. Overlapping Sets For sets A and B drawn in this figure, notice the overlapping area shared by the two circles. This area represents the elements that are in the intersection of set A and set B.

8. Chapter 2 Section 3 - Slide 8 Copyright © 2009 Pearson Education, Inc. Complement of a Set The set known as the complement contains all the elements of the universal set which are not listed in the given subset. Symbol: A ´

9. Chapter 2 Section 3 - Slide 9 Copyright © 2009 Pearson Education, Inc. Intersection The intersection of two given sets contains only those elements common to both of those sets. Symbol:

10. Chapter 2 Section 3 - Slide 10 Copyright © 2009 Pearson Education, Inc. Union The union of two given sets contains all of the elements for both of those sets. The union “unites”, that is, it brings together everything into one set. Symbol:

11. Chapter 2 Section 3 - Slide 11 Copyright © 2009 Pearson Education, Inc. Subsets When every element of B is also an element of A. Circle B is completely inside circle A.

12. Chapter 2 Section 3 - Slide 12 Copyright © 2009 Pearson Education, Inc. Equal Sets When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A. Both sets are drawn as one circle.

13. Chapter 2 Section 3 - Slide 13 Copyright © 2009 Pearson Education, Inc. The Meaning of and and or and is generally interpreted to mean intersection A B = { x | x A and x B } or is generally interpreted to mean union A B = { x | x A or x B }

14. Chapter 2 Section 3 - Slide 14 Copyright © 2009 Pearson Education, Inc. The Relationship Between n(A B), n(A), n(B), n(A B) To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements common to both sets. n(A B) = n(A) + n(B) – n(A B)

15. Chapter 2 Section 3 - Slide 15 Copyright © 2009 Pearson Education, Inc. Difference of Two Sets The difference of two sets A and B symbolized by A – B, is the set of elements that belong to set A but not to set B. Region 1 represents the difference of the two sets. BA U IIIIII IV