open and closed intervals have the same cardinality
The sets of real numbers , , , and all have the same cardinality.
We give two proofs of this proposition.
Define a map by . The map is strictly increasing, hence injective. Moreover, the image of is contained in the interval , so the maps and obtained from by restricting the codomain are both injective. Since the inclusions into are also injective, the Cantor-Schröder-Bernstein theorem (http://planetmath.org/SchroederBernsteinTheorem) can be used to construct bijections and . Finally, the map defined by is a bijection.
Since having the same cardinality is an equivalence relation, all four intervals have the same cardinality. ∎
The reasoning above can be extended to show that any two arbitrary intervals in have the same cardinality.
|Title||open and closed intervals have the same cardinality|
|Date of creation||2013-03-22 15:43:32|
|Last modified on||2013-03-22 15:43:32|
|Last modified by||mps (409)|