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 Venn diagrams Set operations such as complement, intersection, and union Equality of 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. Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 15 Section 4 Venn Diagrams with Three Sets And Verification of Equality of Sets

16. Chapter 2 Section 3 - Slide 16 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets Determine the elements that are common to all three sets and place in region V, A  B  C.

17. Chapter 2 Section 3 - Slide 17 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements for region II. Find the elements in A  B. The elements in this set belong in regions II and V. Place the elements in the set A  B that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner.

18. Chapter 2 Section 3 - Slide 18 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner.

19. Chapter 2 Section 3 - Slide 19 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets (continued) Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII. U A B C V I III VII VI IV VIII II

20. Chapter 2 Section 3 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example: Constructing a Venn diagram for Three Sets Construct a Venn diagram illustrating the following sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8} Solution: Find the intersection of all three sets and place in region V, A  B  C = {5}.

21. Chapter 2 Section 3 - Slide 21 Copyright © 2009 Pearson Education, Inc. Example: Constructing a Venn diagram for Three Sets (continued) Determine the intersection of sets A and B. A  B = {2, 5} Element 5 has already been placed in region V, so 2 must be placed in region II. Now determine the numbers that go into region IV. A  C = {1, 5, 8} Since 5 has been placed in region V, place 1 and 8 in region IV.

22. Chapter 2 Section 3 - Slide 22 Copyright © 2009 Pearson Education, Inc. Example: Constructing a Venn diagram for Three Sets (continued) Now determine the numbers that go in region VI. B  C = {5} There are no new numbers to be placed in region VI. Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III, in which we must write 4. Using set C completes region VII, in which we must write 3.

23. Chapter 2 Section 3 - Slide 23 Copyright © 2009 Pearson Education, Inc. Example: Constructing a Venn diagram for Three Sets (continued) Now place the remaining elements in U (6 and 7) in region VIII. The Venn diagram is then completed.

24. Chapter 2 Section 3 - Slide 24 Copyright © 2009 Pearson Education, Inc. Verification of Equality of Sets To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams. If both statements represent the same regions of the Venn Diagram, then the statements are true for all sets A and B.

25. Chapter 2 Section 3 - Slide 25 Copyright © 2009 Pearson Education, Inc. Example: Equality of Sets Determine whether (A  B) ´ = A ´  B ´ for all sets A and B.

26. Chapter 2 Section 3 - Slide 26 Copyright © 2009 Pearson Education, Inc. Solution Draw a Venn diagram with two sets A and B and label each region. BA U IIIIII IV Find (A  B) ´.Find A ´  B ´. SetRegions AI, II BII, III A BA B II (A  B)´(A  B)´ I, III, IV SetRegions A´A´ III, IV B´B´ I, IV A´  B´A´  B´ I, III, IV

27. Chapter 2 Section 3 - Slide 27 Copyright © 2009 Pearson Education, Inc. Solution Both statements are represented by the same regions, I, III, and IV, of the Venn diagram. Thus, (A  B) ´ = A ´  B ´ for all sets A and B. SetRegions AI, II BII, III A  BA  B II (A  B)´(A  B)´ I, III, IV SetRegions A´A´ III, IV B´B´ I, IV A ´  B ´ I, III, IV

28. Chapter 2 Section 3 - Slide 28 Copyright © 2009 Pearson Education, Inc. De Morgan’s Laws A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms. (A  B) ´ = A ´  B ´ (A  B) ´ = A ´  B ´