Example:
Show that all of the following sets are uncountable:
- The open interval (-1, 1) is uncountable
- Any open interval (a, b) is uncountable
- 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
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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