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

1 Answer
Jun 10, 2017

Answer:

#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.