Set of Finite Strings is Countably Infinite

Theorem

Let $\Sigma$ be a finite alphabet.

Let $\Sigma^*$ be the set of finite strings of $\Sigma$.

Then $\Sigma^*$ is a countably infinite set.

Proof

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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