## 2.2. Uncountable Infinity

### Examples 2.2.2:

Show that all of the following sets are uncountable:

Recall that the set (0, 1) is uncountable, as proved before. Then:
- The open interval (-1, 1) is uncountable
- Any open interval (a, b) is uncountable
- The set of all real numbers R is uncountable

1. Define the function

*f(x) = 2x - 1*from (0, 1) to (-1, 1)

2. A similar proof can show that any open interval *(a, b)* is
uncountable. What is the appropriate bijection (try a linear function that
maps 0 to *a* and 1 to *b*) ?

3. Define a function

*f(x) = x - / 2*

*g(x) = tan(x)*

**R**. But then the composition of the two function will be a bijection from (0, 1) to

**R**, and hence both sets must have the same cardinality.