Power Set of Set of Finite Strings is Uncountable
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Theorem
Let $\Sigma$ be a finite alphabet.
Let $\Sigma^*$ be the set of finite strings of $\Sigma$.
Let $\powerset {\Sigma^*}$ be the power set of $\Sigma^*$
Then $\powerset {\Sigma^*}$ is an uncountable set.
Proof
From Set of Finite Strings is Countably Infinite, $\Sigma^*$ is a countably infinite set.
The result follows from Power Set of Countably Infinite Set is Uncountable.
$\blacksquare$
Sources
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.4$ Set Notation: Infinite sets