2.6 Infinite Sets 89 Countable Sets In his work with infinite sets, Cantor developed ideas on how to determine the cardinal number of an infinite set. He called the cardinal number of infinite sets “transfinite cardinal numbers” or “transfinite powers.” He defined a set as countable if it is finite or if it can be placed in a one-to-one correspondence with the set of counting numbers. All infinite sets that can be placed in a one-to-one correspondence with the set of counting numbers have cardinal number aleph-null, symbolized 0ℵ (the first Hebrew letter, aleph, with a zero subscript, read “null”). Example 3 An Infinite Set Show that the set − n {1, 4, 7, 10, , 3 2, } … … is an infinite set. Solution First create a proper subset of the given set by removing the first element from the set. Then establish a one-to-one correspondence. In this correspondence, notice that 1 corresponds with 4, 4 corresponds with 7, 7 corresponds with 10, and so on. Note that the element in the subset is always 3 greater than the corresponding element in the given set. Therefore, to determine the expression that corresponds to n3 2, − we need to add 3 to n3 2. − This will give us n n 3 2 3 3 1. − + = + − ↓ ↓ ↓ ↓ ↓ + n n {1, 4, 7, 10, ..., 3 2, ...} {4, 7, 10, 13, ..., 3 1, ...} Therefore, the given set is an infinite set. 7 Now try Exercise 7 Learning Catalytics Keyword: Angel-SOM-2.6 (See Preface for additional details.) Example 4 The Cardinal Number of the Set of Even Numbers Show that the set of even counting numbers has cardinal number . 0ℵ Solution In Example 2, we showed that the set of even counting numbers is infinite by setting up a one-to-one correspondence between the set and a proper subset of itself. Now we will show that it is countable and has cardinality 0ℵ by setting up a one-to-one correspondence between the set of counting numbers and the set of even counting numbers. = … … ↓ ↓ ↓ ↓ ↓ = … … N n E n Counting numbers: {1, 2, 3, 4, , , } Even counting numbers: {2, 4, 6, 8, , 2 , } For each number n in the set of counting numbers, its corresponding number is n2 . Since we determined a one-to-one correspondence between the set of counting numbers and the set of even counting numbers, the set of even counting numbers is countable. Thus, the cardinal number of the set of even counting numbers is ;0ℵ that is, n E( ) . 0 =ℵ As we mentioned earlier, the set of even counting numbers is an infinite set, since it can be placed in a one-to-one correspondence with a proper subset of itself. Therefore, the set of even counting numbers is both infinite and countable. 7 Now try Exercise 13 Definition: Cardinal Number of Infinite Sets Any set that can be placed in a one-to-one correspondence with the set of counting numbers has cardinal number (or cardinality) 0ℵ and is infinite and is countable.
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