## Example 2.2.7: A Hierarchy of Infinity - Cardinal Numbers

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.

*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(*and**A**) = c*card(*and**B**) = d*card(***A**) card(**B**)

### 3. Special Cardinalities

The cardinality of the real numbers is called the**cardinality (or power) of the continuum**, and is denoted by

*c = card(*.**R**)

*= card(***N**)

### 4. Adding Cardinal Numbers

Let*c*and

*d*be two cardinal numbers and take sets

**A**and

**B**with

*card(*and

**A**) = c*card(*. Define the sets

**B**) = d**A'**= {(a, 0) : a A}**B'**= {(b, 1) : b B}

*c*and

*d*is defined as

*c + d = card(***A'****B'**)

**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(*and

**A**) = 3*card(*. However,

**B**) = 2*card(***A****B**) = 3

**A**

**B**= {1,2,3}. Using the above definition, we get:

**A'**= { (1,0), (2,0), (3,0)}**B'**= { (1,1), (2,1) }

**A'****B'**= {(1,0), (2,0), (3,0), (1,1), (2,1)}

*card(***A'****B'**) = 5 = card(**A**) + card(**B**)