88 CHAPTER 2 Sets Note in Example 1 that we showed the pairing of the general terms n n( 1). → + Showing a one-to-one correspondence of infinite sets requires showing the pairing of the general terms in the two infinite sets. In the set of counting numbers, n represents the general term. For any other set of numbers, the general term will be different. The general term in any set should be written in terms of n such that when 1 is substituted for n in the general term, we get the first number in the set; when 2 is substituted for n in the general term, we get the second number in the set; when 6 is substituted for n in the general term, we get the sixth number in the set; and so on. Consider the set {4, 9, 14, 19, }. … Suppose we want to write the general term for this set (or sequence) of numbers. What would the general term be? The numbers differ by 5, so the general term will be of the form n5 plus or minus some number. Substituting 1 for n yields 5(1), or 5. Because the first number in the set is 4, we need to subtract 1 from the 5. Thus, the general term is n5 1. − Note that when n 1, = the value is 5(1) 1− or 4; when n 2, = the value is 5(2) 1− or 9; when n 3, = the value is 5(3) 1− or 14; and so on. Therefore, we write the set of numbers with the general term as … − … n {4, 9, 14, 19, , 5 1, } Now that you are aware of how to determine the general term of a set of numbers, we can do some more problems involving sets. Example 1 The Set of Natural Numbers Show that = … … N n {1, 2, 3, 4, 5, , , } is an infinite set. Solution To show that the set N is infinite, we establish a one-to-one correspondence between the counting numbers and a proper subset of itself. By removing the first element from the set of counting numbers, we get the set {2, 3, 4, 5, 6, }, … which is a proper subset of the set of counting numbers. Now we establish the one-to-one correspondence. = … … ↓↓ ↓ ↓ ↓ ↓ = … + … n n Counting numbers {1, 2, 3, 4, 5, , , } Proper subset {2, 3, 4, 5, 6, , 1, } Note that for any number, n, in the set of counting numbers, its corresponding number in the proper subset is one greater, or n 1. + We have now shown the desired oneto-one correspondence, and thus the set of counting numbers is infinite. 7 Now try Exercise 3 Profile in Mathematics Leopold Kronecker Mathematician Leopold Kronecker (1823–1891), Cantor’s former mentor, ridiculed Cantor’s theories and prevented Cantor from gaining a position at the University of Berlin. Although Cantor’s work on infinite sets is now considered a masterpiece, it generated heated controversy when originally published. Cantor’s claim that the infinite set was unbounded offended the religious views of the time that God had created a complete universe that could not be wholly comprehended by humans. Eventually Cantor was given the recognition due him, but by then the criticism had taken its toll on his health. He had several episodes of intense mental distress and spent his last days in a psychiatric hospital. See the Profile in Mathematics on page 42 for more information on Cantor. Example 2 The Set of Even Numbers Show that the set of even counting numbers … … n {2, 4, 6, 8, , 2 , } is an infinite set. Solution First create a proper subset of the set of even counting numbers by removing the first number from the set. Then establish a one-to-one correspondence. n n Even counting numbers: {2, 4, 6, 8, , 2 , } Proper subset: {4, 6, 8, 10, , 2 2, } … … ↓↓ ↓ ↓ ↓ … + … A one-to-one correspondence exists between the two sets, so the set of even counting numbers is infinite. 7 Now try Exercise 5
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