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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.
Example 1: The Set of Natural
Numbers
Show that N = {1, 2, 3, 4, 5, …, n,…}
is an infinite set.
2.6-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: The Set of Natural
Numbers
Solution
Remove the first element of set N, to
get the proper subset P of the set of
counting numbers
N = {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.
2.6-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: The Set of Natural
Numbers
Solution
We 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,…}
2.6-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try this: p. 86 #4
2.6-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: The Set of Multiples
of Five
Show that N = {5, 10, 15, 20,…,5n,…}
is an infinite set.
2.6-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: The Set of Natural
Numbers
Solution
Given
Set =
{5, 10, 15, 20, 25,…, 5n,…}
Proper
Subset = {10, 15, 20, 25,
30,…,5n+5,…}
Therefore, the given set is infinite.
2.6-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try this #8
2.6-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
•
2.6-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Cardinal Number of Infinite Sets
Any set that can be placed in a oneto-one correspondence with the set of
counting numbers has cardinal
number (or cardinality)ℵ0, and is
infinite and is countable.
2.6-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: The Cardinal Number
of the Set of Odd Numbers
Show the set of odd counting numbers
has cardinality ℵ0.
2.6-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: The Cardinal Number
of the Set of Odd Numbers
Solution
We 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,…}
2.6-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: The Cardinal Number
of the Set of Odd Numbers
Solution
Since there is a one-to-one
correspondence, the odd counting
numbers have cardinality ℵ0; that is
n(O) = ℵ0.
N = {1, 2, 3, 4, 5,…,
n,…}
O = {1, 3, 5, 7, 9,…, 2n–1,…}
2.6-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try This: P. 87 # 16
2.6-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Homework
P. 86 – 87 # 3 – 21 (x3)
2.6-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.