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# Differences

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sav08:countable_set [2008/03/10 16:27]
vkuncak created
sav08:countable_set [2008/03/10 16:29] (current)
vkuncak
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**Theorem:​** if $A, B$ are countable then $A \cup B$ and $A \times B$ are countable, but $2^A$ is not countable. **Theorem:​** if $A, B$ are countable then $A \cup B$ and $A \times B$ are countable, but $2^A$ is not countable.

+Observation:​ The set of all strings over some finite alphabet is countable.  ​
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+Observation:​ The set of real numbers is not countable.
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+Observation:​ If the set $A$ is infinite and the set $B$ has at least two elements, then set of all functions $f : A \to B$ is not countable.