Question

Which of the following statements about the power set of a set are true?

1) May be finite
2) May be countably infinite
3) May be uncountable
4) May be countable

Answer 1

`1`, `3` and `4` are true. `2` is false.

Explanation:

1) May be finite - Yes

If `A` is finite then so is its power set, with cardinality `2^abs(A)`

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2) May be countably infinite - No

If `A` is infinite then it is of cardinality at least that of `NN` and it's power set has cardinality at least `2^abs(NN) = 2^omega`, which is uncountable by Cantor's diagonal argument.

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3) May be uncountable - Yes

If `A = NN`, then `2^A` is uncountable.

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4) May be countable - Yes

If `A` is finite, then `2^A` is also finite and therefore countable.