# Interactive Real Analysis - part of MathCS.org

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## Examples 2.2.2:

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

1. Define the function

• f(x) = 2x - 1 from (0, 1) to (-1, 1)
This is a bijection between those two intervals, and therefore both intervals have the same cardinality.

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
Then this function is a bijection between the open intervals (0, 1) and (- / 2, / 2). Next, take the function
• g(x) = tan(x)
This function is a bijection between (- / 2, / 2) and 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.
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