copyright 2013, 2010, 2007, pearson, education, inc. section 2.6 infinite sets
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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Infinite Set An infinite set is a set that can be placed in a one-to-one correspondence with a proper subset of itselfTRANSCRIPT
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 2.6
Infinite Sets
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Infinite Sets
2.6-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Infinite Set
An infinite set is a set that can be placed in a one-to-one correspondence with a proper subset of itself.
2.6-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Show that N = {1, 2, 3, 4, 5, …, n,…} is an infinite set.
Example 1: The Set of Natural Numbers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionRemove the first element of set N, to get the proper subset P of the set of counting numbersN = {1, 2, 3, 4, 5,…, n,…}
P = {2, 3, 4, 5, 6,…, n + 1,…}For any number n in N, its corresponding number in P is n + 1.
Example 1: The Set of Natural Numbers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionWe have shown the desired one-to-one correspondence, therefore the set of counting numbers is infinite.
N = {1, 2, 3, 4, 5,…, n,…}
P = {2, 3, 4, 5, 6,…, n + 1,…}
Example 1: The Set of Natural Numbers
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Show that N = {5, 10, 15, 20,…,5n,…} is an infinite set.
Example 3: The Set of Multiples of Five
2.6-7
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SolutionGivenSet = {5, 10, 15, 20, 25,…, 5n,…}
ProperSubset = {10, 15, 20, 25, 30,…,5n+5,…}
Therefore, the given set is infinite.
Example 3: The Set of Natural Numbers
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Countable Sets• A set is 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 a set of counting numbers have cardinal number aleph-null, symbolized ℵ0.
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Cardinal Number of Infinite SetsAny set that can be placed in a one-to-one correspondence with the set of counting numbers has cardinal number (or cardinality)ℵ0, and is infinite and is countable.
2.6-10
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Show the set of odd counting numbers has cardinality ℵ0.
Example 5: The Cardinal Number of the Set of Odd Numbers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionWe need to show a one-to-one correspondence between the set of counting numbers and the set of odd counting numbers.
N = {1, 2, 3, 4, 5,…, n,…}
O = {1, 3, 5, 7, 9,…, 2n–1,…}
Example 5: The Cardinal Number of the Set of Odd Numbers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionSince there is a one-to-one correspondence, the odd counting numbers have cardinality ℵ0; that isn(O) = ℵ0.
N = {1, 2, 3, 4, 5,…, n,…}
O = {1, 3, 5, 7, 9,…, 2n–1,…}
Example 5: The Cardinal Number of the Set of Odd Numbers
2.6-13