Chapter 2

The Basic Concepts of Set Theory

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Chapter 2: The Basic Concepts of Set Theory

2.1 Symbols and Terminology

2.2 Venn Diagrams and Subsets

2.3 Set Operations and Cartesian Products

2.4 Surveys and Cardinal Numbers

2.5 Infinite Sets and Their Cardinalities

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Chapter 1

Section 2-5Infinite Sets and Their Cardinalities

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Infinite Sets and Their Cardinalities

• One-to-One Correspondence and Equivalent Sets

• The Cardinal Number (Aleph-Null)

• Infinite Sets

• Sets That Are Not Countable

0

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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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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}

Example: One-to-One Correspondence

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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

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 (aleph-null).

0

0

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To show that another set has cardinal number we show that it is equivalent to the set of counting numbers.

{1, 2, 3, 4, …, n, …}

{2, 4, 6, 8, …,2n, …}

Example: Showing That {2, 4, 6, 8,…} Has Cardinal Number 0

0 ,

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Infinite Sets

A set is infinite if it can be placed in a one-to-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

Whole Numbers

Integers

Rational Numbers

Irrational Numbers c

Real Numbers c

0

0

0

0