Example:
When dealing with cardinal numbers, one can establish the following rules and
definitions:
- Definition of a cardinal number
- Comparing cardinal numbers
- The power of the continuum and the cardinality aleph null
- Addition of two cardinal numbers
1. Definition of Cardinal Number
- Two sets A and B are called equivalent if there exists a
bijection between A and B. The two sets are said to have the same
cardinality, or power.
- The cardinality of a set A is denoted by
card(A).
- The cardinal numbers of two sets are equal if the sets are
equivalent.
Note that for finite sets the cardinality is just the number of elements in
that set. That is, card({a, b, 2} ) = card({1, 2, 9}) = 3. However, the
concept of cardinality also applies to infinite sets.
2. Comparing Cardinal Numbers
- A cardinal number c is less than or equal to another cardinal
number d if there exist two sets A and B with
card(A) = c and card(B) = d and
card(A)
card(B)
Note that according to this definition, we have that the cardinality of the
natural numbers is strictly less than the cardinality of the real numbers. In
fact, those cardinalities have a special name:
3. Special Cardinalities
The cardinality of the real numbers is called the cardinality (or power)
of the continuum, and is denoted by
The cardinality of the natural number is called aleph null and is denoted by
= card(N)
4. Adding Cardinal Numbers
Let c and d be two cardinal numbers and take sets A
and B with card(A) = c and card(B) = d.
Define the sets
- A' = {(a, 0) : a
A}
- B' = {(b, 1) : b B}
Then the sum of the two cardinal number c and d is defined as
- c + d = card(A'
B')
Note that the reason for defining the sets A' and B' is to make
sure that the resulting sets are disjoint. If the two sets A and B
are disjoint from the outset, one could define the sum of the cardinal numbers
as the cardinality of the union of the original sets.
Example:
Let A = {1,2,3} and B = {1,2}. Then card(A) = 3
and card(B) = 2. However,
- card(A B) = 3
because A B = {1,2,3}.
Using the above definition, we get:
- A' = { (1,0), (2,0), (3,0)}
- B' = { (1,1), (2,1) }
therefore
- A' B' =
{(1,0), (2,0), (3,0), (1,1), (2,1)}
and therefore, as one would think:
- card(A' B')
= 5 = card(A) + card(B)
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