## 2.2. Uncountable Infinity

### Examples 2.2.8:

We want to add or subtract the following cardinal numbers:

*card(N) + card(N) = card(N)**card(N) - card(N) = undefined**card(R) + card(N) = card(R)**card(R) + card(R) = card(R)*

### 1. *card(N) + card(N) = card(N)*

According to the definition, this is the same as the cardinality of
**A**

**B**, where

**A**and

**B**are both countable, disjoint sets . But the countable union of countable sets is again countable. Hence,

*card(*, so that

**A****B**) = card(**N**)*card(***N**) + card(**N**) = card(**N**)

*= aleph null = card(*

**N**))- + =

### 2. *card(N) - card(N) = undefined*

Although this has not been properly defined, one could say that this should
be the same as the cardinality of **A**\

**B**, where

**A**and

**B**are both countable sets and

**B**is a subset of

**A**. This creates problems, however, as the following examples show:

**A**=**B**=**N**. Then*card(***A**\**B**) = card(**0**) = 0**A**= all integers,**B**= even integers. Then*card(***A**\**B**) = card(odd integers) = card(**N**)

*card(*is undefined.**N**) - card(**N**)

- - is undefined

### 3. *card(R) + card(N) = card(R)*

According to the definition, this is the same as the cardinality of
**A**

**B**, where

**A**is uncountable and

**B**is countable and

**A**and

**B**are disjoint. We know that every subset of a countable set is countable or finite. Since

**A**is a subset of

**A**

**B**, the set

**A**

**B**can not be countable. Hence, it must be uncountable. We would therefore guess that

*card(***R**) + card(**N**) = card(**R**)

*c + = c*

**A**

**B**can not be strictly larger that the cardinality of

**A**to establish this. That, however, is left as an exercise.

### 4. *card(R) + card(R) = card(R)*

This should be the same as the cardinality of
**A**

**B**, where both

**A**and

**B**are uncountable and disjoint. It is easy to find a one-to-one function from

**A**to

**A**

**B**, so that

*card(*. But then

**A**) card(**A****B**)*card(*is again uncountable, so that we would guess that

**A****B**)*card(***R**) + card(**R**) = card(**R**)

*c + c = c*

*card(*This is true, and left as an exercise.

**A****B**) card(**A**)