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Infinite Sets and Cardinalities

Correspondence, 0 , Sets that are not countable

When each element of the first set can be paired with exactly one element of the second set

Set A= {1, 2, 3, 4,}

Set B= {10, 11, 12, 13} Sets with one-to-one correspondence are said

to be equivalent (different than being equal) A~B

Empty sets ( There is nothing to pair!

One-To-One Correspondence

{Example; non-equivalent sets

{3, 7, 11}

{2, 4}

{Cardinal number notation; n(A)

If two non-empty sets have the same cardinal number, they have a one-to-one correspondence

The most common infinite set we refer to is the set of counting numbers; {1, 2, 3, 4, 5, …}

The way we symbolize the cardinality of the set is 0 ( read; aleph-null)

0

Intuitively, this seems incorrect. Counting numbers should have one less element than whole numbers since they start at 0 instead of 1, right? (Galileo’s Paradox)

Since they are infinite, however, we have a one-to-one ratio

I0

Counting Numbers { 1, 2, 3, 4, 5, 6 ….. n }

Whole Numbers; { 0, 1, 2, 3, 4, 5, …. n-1}

A Proper Subset of a set has least one less element than that set

P= {2, 3, 6, 9} A PROPER subset would be {3, 6, 9}

Counting Numbers are a proper Subset of Whole numbers

(counting numbers are all the same numbers, excluding 0)

Back to Proper Subsets

{Definition of an Infinite set

This fact gives us a new definition for an infinite set;A set is infinite if it can be placed in a one-to-one correspondence with a proper subset of itself

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• The set of Integers; {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}How can we show a one-to-one correspondence?

Example; Show the set of integers is an infinite set

{1, 2, 3, 4, 5, 6, 7, …}

{0, 1, -1, 2, -2, 3, -3, ….}

Subset of the set of integers

A set is countable if It is Finite or Has a cardinal number 0

Countable Sets

A set is not countable if it does not have a cardinality of 0

We cannot set up a one-to-one correspondence between a non-countable set and the set of counting numbers

Sets that are not countable

The set of real numbers are all numbers that can be written as decimals.

Because there is an infinite continuum from, say, 1 to 2, you cannot set up a one-to-one correspondence

1, 1.1, 1.01, 1.001…. 1.12, 1.13, 1.14 You can keep adding more and more

numbers between 1 & 2 In between every number, there is an

infinite amount of numbers

Real numbers; not countable

Any set that is not countable is considered a continuum,

Notation;

Real numbers; not countable

{Summary

Infinite Set Cardinal Number

Natural/ Counting #s 0

Whole Numbers 0

Integers 0

Rational Numbers 0

Irrational Numbers

Real Numbers