90 CHAPTER 2 Sets We have shown that both the odd and the even counting numbers have cardinality . 0ℵ Merging the odd counting numbers with the even counting numbers gives the set of counting numbers, and we may reason that 0 0 0 ℵ +ℵ =ℵ This result may seem strange, but it is true. What could such a statement mean? Well, consider a hotel with infinitely many rooms. If all the rooms are occupied, the hotel is, of course, full. If more guests appear, wanting accommodations, will they be turned away? The answer is no, for if the room clerk were to reassign each guest to a new room with a room number twice that of the present room, all the odd-numbered rooms would become unoccupied and there would be space for infinitely many more guests! Cantor showed that there are different orders of infinity. Sets that are countable and have cardinal number 0ℵ are the lowest order of infinity. Cantor showed that the set of integers and the set of rational numbers (fractions of the form p q/ , where q 0 ≠ ) are infinite sets with cardinality . 0ℵ He also showed that the set of real numbers (discussed in Chapter 5) could not be placed in a one-to-one correspondence with the set of counting numbers and that they have a higher order of infinity. Example 5 The Cardinal Number of the Set of Odd Numbers Show that the set of odd counting numbers has cardinality . 0ℵ Solution To show that the set of odd counting numbers has cardinality , 0ℵ we need to show a one-to-one correspondence between the set of counting numbers and the set of odd counting numbers. = … … ↓ ↓ ↓ ↓ ↓ ↓ = … − … N n O n Counting numbers: {1, 2, 3, 4, 5, , , } Odd counting numbers: {1, 3, 5, 7, 9, , 2 1, } Since there is a one-to-one correspondence, the odd counting numbers have cardinality ;0ℵ that is, n O( ) . 0 =ℵ 7 Now try Exercise 17 m … where there’s always room for one more… Exercises Warm Up Exercises In Exercises 1–2, fill in the blank with an appropriate word, phrase, or symbol(s). 1. A set that can be placed in a one-to-one correspondence with a proper subset of itself is called a(n) _______ set. Infinite 2. A set that is finite or can be placed in a one-to-one correspondence with the set of counting numbers is called a(n) _______ set. Countable Practice the Skills In Exercises 3–12, show that the set is infinite by placing it in a one-to-one correspondence with a proper subset of itself. Be sure to show the pairing of the general terms in the sets. 3. + … … n {5, 6, 7, 8, 9, , 4, } * 4. … + … n {30, 31, 32, 33, 34, , 29, } * 5. … + … n {6, 8, 10, 12, 14, , 2 4, } * 6. … + … n {5, 8, 11, 14, 17, , 3 2, } * 7. … … + n {5, 7, 9, 11, 13, , 2 3, } * 8. + … … n {5, 9, 13, 17, 21, , 4 1, } * 9. … … {, , , , , , , } n 1 2 1 4 1 6 1 8 1 10 1 2 * 10. … … {1, , , , , , , } n 1 2 1 3 1 4 1 5 1 * 11. … … + {, , , , , , , } n 4 11 5 11 6 11 7 11 8 11 3 11 * 12. + … … {, , , , , , , } n 6 13 7 13 8 13 9 13 10 13 5 13 * In Exercises 13–22, show that the set has cardinal number 0ℵ by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairing of the general terms in the sets. 13. … {3, 6, 9, 12, 15, } * 14. … {20, 21, 22, 23, 24, } * 15. … {4, 6, 8, 10, 12, } * 16. … {0, 2, 4, 6, 8, } * SECTION 2.6 Instructor Resources for Section 2.6 in MyLab Math • Objective-Level Videos 2.6 • PowerPoint Lecture Slides 2.6 • MyLab Exercises and Assignments 2.6 • Chapter 2 Projects *See Instructor Answer Appendix
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