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Let Σ be a finite non-empty alphabet and let 2^Σ* be the power set of Σ*. Which one of the following is TRUE ?

1. Both 2^Σ* and Σ* are countable
2. 2^Σ* is countable and Σ* is uncountable
3. 2^Σ* is uncountable and Σ* is countable
4. Both 2^Σ* and Σ* are uncountable

Correct answer: 2^Σ* is uncountable and Σ* is countable

Solution

The set Σ* consists of all finite strings over a finite alphabet Σ, making it countable since it can be listed. However, the power set 2^Σ*, which includes all possible subsets of Σ*, is uncountable, as it has a greater cardinality than Σ* itself.

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