Venn Diagrams and the Equality of Set Expressions Two sets are equal if and only if they each represent the same region(s) on a Venn diagratm. Venn diagrams can be used to verify each of the following properties. CHAPTER 2 Reviea Exerses For all sets A, B, and C: 101 See Example 6 and Check Youue Progress 6 on page 71, and then try Exercises 45 to 48 on page 102. Commutative Properties ANB=BNA AUB=BU A Associative Properties (AN B) N C = AN (B n C) An(B U C) = (A N B) U (A n C) (AU B) U C = A U (B U C) AU (Bnc)= (AU B)n (AU C) Distributive Properties 2.4 Applications of Sets Applications Many counting problems that arise in applications involving surveys can be solved by using sets and Venn diagrams. The Inclusion-Exclusion Formula For all finite sets A and B, See Examples 1 and 2 on page 81, and then try Exercises 53 and 54 on page 102. n(A U B) = n(A) + n(B) – n(A N B) See Examples 3 and 4 on page 83, and then 2.5 Infinite Sets Exercises 55 and 56 on page 103. try One-to-One Correspondence and Equivalent Sets Two sets A and B re equivalent, denoted by A ~ B, if and only if A and B can be placed See Examples 1 and 3 on pages 90 and 92, and then try Exercises 57 to 60 on page 103. in a one-to-one correspondence. Infinite Set A set is an infinite set if it can be placed in a one-to-one correspondence with a proper subset of itself. See Example 2 on page 91, and then try Exercises 61 and 62 on page 103. CHAPTER 2 REVIEW EXERCISES In Exercises 11 and 12, state whether each of the fol- lowing sets are equal, equivalent, both, or neither. 11. {2, 4, 6, 8}, {x\x E N and x < 5} - In Exercises 1 to 6, use the roster method to represent each set. 1. The set of months of the year with a name that starts 12. {8, 9}, the set of single digit whole numbers greater than 7 with the letter J 2. The set of states in the United States that do not share a common border with another state I In Exercises 13 to 16, determine whether the statement is A.A2 3. The set of whole numbers less than 8 64 true or false. 13. {3} E {1,2, 3, 4} 4. The set of integers that satisfy x 14. -11 EI 15. {a, b, c} ~ {1, 5, 9} 5. The set of natural numbers that satisfy x + 3 <7 6. The set of counting numbers larger than -3 and less than or equal to 6 In Exercises 17 to 24, let U = {2, 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14, 16}. Find each of the following. 16. The set of small numbers is a well-defined set %3D - In Exercises 7 to 10, use set-builder notation to write 18. AUB each set. 17. ANB 20. BUC' 7. The set of integers greater than -6 19. A' N C 22. (A U C)' N B' 8. {April, June, September, November} 21. A U (B N C) 24. (AUBUC)' 9. {Kansas, Kentucky} 23. (A N B')' 10. {1, 8, 27, 64, 125} froper given storlemen