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

Jun 10, 2017

$1$, $3$ 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 ${2}^{\left\mid A \right\mid}$

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2) May be countably infinite - No

If $A$ is infinite then it is of cardinality at least that of $\mathbb{N}$ and it's power set has cardinality at least ${2}^{\left\mid \mathbb{N} \right\mid} = {2}^{\omega}$, which is uncountable by Cantor's diagonal argument.

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3) May be uncountable - Yes

If $A = \mathbb{N}$, 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.