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Lets say the alphabet is Σ=∅,what are the possible languages of this alphabet?

According to my definitions:

  • I know that an alphabet is a finite set of symbols Σ
  • I know words is a set of all finite symbols consisting of letters from the alphabet (Σ*) and an empty string part of all words ε ∈ Σ*
  • I know that a language is a subset of words and an empty set is a subset of the words ∅ ⊆ Σ*

All this is very confusing to me, and I am very unsure about what languages there are when we have an empty set for the alphabet.

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  • $\begingroup$ It's important to understand the difference between a word (which is a sequence of symbols), a language, which is a set of words, and a set of languages. Those are three different things, which are not interchangeable. Also, always keep in mind the difference between the notions of element and subset. $\endgroup$
    – rici
    Sep 3, 2022 at 20:07

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Actually your list of properties already solves the problem. As $\varepsilon$ is an element of $\Sigma^*$ and we cannot have any longer word due to the lack of symbols, we have $\varnothing^*=\{\varepsilon\}$.

A language is a subset of $\varnothing^*$, and as $\varnothing^*$ has only one element, there are two possible languages.

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  • $\begingroup$ So does that mean that the language (subsets of ∅) is just {𝜀} and ∅? $\endgroup$
    – MohG
    Sep 3, 2022 at 13:32

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