Is P(Σ∗) countable for any finite Σ?

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Expert Solution

Proof:

The length of a string is the number of images in the string. For instance, the length of 69302040 is 8, also, the length of +6949302 is 7.
We can compose Σ∗ as the association of Σ for n = 0, 1, 2, . . . , where Σ^n is the set of all strings over Σ having length l. Each Σ^n has a limited size, as is countable. Along these lines, Σ∗is the association of countably numerous countable sets.

The above proposition stays substantial regardless of whether we permit Σ to be countably infinite. A language over Σ is a subset of Σ∗. For instance, the language of English comprises correctly of those strings (over the Roman letter set) that has an elucidation in English. In this way, 'Sir, I'm Jhon!' is in the English language, though 'I Jhon, Sir am!' isn't in the English language. Also, +0.12.345 is anything but a numeric string (i.e., not in the language of genuine numbers). The set of all languages over Σ is exactly the power set P(Σ∗) of Σ∗.

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