4. If two non-empty sets have
the same cardinal
number, they have a one-to-
one correspondence
{ Cardinal number notation; n(A)
5.
6. 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
Counting Numbers { 1, 2, 3, 4, 5, 6 ….. n }
Whole Numbers; { 0, 1, 2, 3, 4, 5, …. n-1}
7. 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
8. 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
{ Definition of an Infinite set
9. • The set of Integers; {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}
How can we show a one-to-one correspondence?
{1, 2, 3, 4, 5, 6, 7, …} Subset of the set of
integers
{0, 1, -1, 2, -2, 3, -3, ….}
{ Example; Show the set of integers is an infinite set
12. 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
