Transcript Mathematical Ideas - Millersville University of Pennsylvania

Chapter 2
The Basic
Concepts of
Set Theory
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 2: The Basic Concepts of
Set Theory
2.1
2.2
2.3
2.4
2.5
Symbols and Terminology
Venn Diagrams and Subsets
Set Operations and Cartesian Products
Surveys and Cardinal Numbers
Infinite Sets and Their Cardinalities
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2-5-2
Chapter 1
Section 2-5
Infinite Sets and Their Cardinalities
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2-5-3
Infinite Sets and Their Cardinalities
• One-to-One Correspondence and Equivalent
Sets
• The Cardinal Number 0 (Aleph-Null)
• Infinite Sets
• Sets That Are Not Countable
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2-5-4
One-to-One Correspondence and
Equivalent Sets
A one-to-one correspondence between two
sets is a pairing where each element of one set
is paired with exactly one element of the
second set and each element of the second set
is paired with exactly one element of the first
set.
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Example: One-to-One Correspondence
For sets {a, b, c, d} and {3, 7, 9, 11} a pairing
to demonstrate one-to-one correspondence
could be
{a, b, c, d}
{3, 7, 9, 11}
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Equivalent Sets
Two sets, A and B, which may be put in a
one-to-one correspondence are said to be
equivalent, written A ~ B.
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The Cardinal Number
0
The basic set used in discussing infinite sets
is the set of counting numbers, {1, 2, 3, …}.
The set of counting numbers is said to have
the infinite cardinal number 0 (aleph-null).
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Example: Showing That {2, 4, 6, 8,…}
Has Cardinal Number 0
To show that another set has cardinal number 0 ,
we show that it is equivalent to the set of
counting numbers.
{1, 2, 3, 4, …, n, …}
{2, 4, 6, 8, …,2n, …}
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Infinite Sets
A set is infinite if it can be placed in a oneto-one correspondence with a proper subset
of itself.
The whole numbers, integers, and rational
numbers have cardinal number 0 .
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Countable Sets
A set is countable if it is finite or if it has
cardinal number 0 .
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Sets That Are Not Countable
The real numbers and irrational numbers are
not countable and are said to have cardinal
number c (for continuum).
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Cardinal Numbers of Infinite Sets
Infinite Set
Cardinal Number
Natural Numbers
0
Whole Numbers
0
Integers
0
Rational Numbers
0
Irrational Numbers
c
Real Numbers
c
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2-5-13