2.6 Infinite Sets 91 17. … {2, 5, 8, 11, 14, } * 18. … {7, 11, 15, 19, 23, } * 19. … {, , , , , } 1 3 1 6 1 9 1 12 1 15 * 20. … {, , , , } 1 2 1 4 1 6 1 8 * 21. … {, , , , , } 1 3 1 4 1 5 1 6 1 7 * 22. … {, , , , , } 1 2 2 3 3 4 4 5 5 6 * Challenge Problems/Group Activities In Exercises 23–26, show that the set has cardinality 0ℵ by establishing a one-to-one correspondence between the set of counting numbers and the given set. 23. … {1, 4, 9, 16, 25, } * 24. … {2, 4, 8, 16, 32, } * 25. … {3, 9, 27, 81, 243, } * 26. … {, , , , , } 1 3 1 6 1 12 1 24 1 48 * Recreational Mathematics In Exercises 27–31, insert the symbol <, >, or = in the shaded area to make a true statement. 27. ℵ ℵ +ℵ = 0 0 0 28. ℵ ℵ +ℵ = 2 0 0 0 29. ℵ ℵ = 2 0 0 30. 5 3 0 0 ℵ + ℵ + = 31. ℵ = n N( ) 0 32. Zeno's Paradox There are a number of paradoxes (a statement that appears to be true and false at the same time) associated with infinite sets and the concept of infinity. One of these, called Zeno’s paradox, is named after the mathematician Zeno, born about 496 b.c. in Italy. According to Zeno’s paradox, suppose Achilles starts out 1 meter behind a tortoise. Also, suppose Achilles walks 10 times as fast as the tortoise crawls. When Achilles reaches the point where the tortoise started, the tortoise is 1/10 of a meter ahead of Achilles; when Achilles reaches the point where the tortoise was 1/10 of a meter ahead, the tortoise is now 1/100 of a meter ahead; and so on. According to Zeno’s paradox, Achilles gets closer and closer to the tortoise but never catches up to the tortoise. a) Do you believe the reasoning process is sound? If not, explain why not. Answers will vary. b) In actuality, if this situation were real, would Achilles ever pass the tortoise? No Research Activities 33. Rational Numbers Do research and write a report explaining how Cantor proved that the set of rational numbers has cardinal number .0ℵ 34. Real Numbers Do research and write a report explaining how it can be shown that the set of real numbers do not have cardinal number .0ℵ *See Instructor Answer Appendix ■ ■ ■ ■ ■ Important Facts and Concepts Examples and Discussion Section 2.1 Methods Used to Indicate a Set Description Roster form Set-builder notation Symbol Meaning ∈ is an element of ∉ is not an element of n A( ) number of elements in set A ∅ or {} the empty set U the universal set Example 2, page 43 Examples 3, 4, 6, 8, pages 43–54 Examples 5– 6, page 44 Examples 5–7, 10 –12, page 44– 45, 46– 48 Section 2.2 Symbol Meaning # is a subset of Ü is not a subset of , is a proper subset of ÷ is not a proper subset of Number of distinct subsets of a finite set with n elements is 2 .n Number of distinct proper subsets of a finite set with n elements is − 2 1. n Examples 1 and 3, pages 52–54 Examples 4 and 5, page 55 Example 4, page 55 Summary CHAPTER 2
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