The statement given is "P(A) is a countable set for some A." where A is any set and P(A) is a power set of A.

Now I know that P(A) can never be countably infinite for any A but P(A) can be finite for some A.

May be since English is not my first language I'm struggling to answer whether the given statement is true or false.

I tried rewriting that statement as follows:

"P(A) is a finite or countably infinite set for some A."

Is it safe now to declare this statement to be true because the 'or' inside the statement can clear our way towards finiteness?

I wrote down contrapositive statement too:

"P(A) is neither finite nor countably infinite for all A."

This version seems true since P(A) can be uncountable.

So I think the given statement is true.

PS: I was taught that the word countable is same as 'finite or countably infinite'.

  • $\begingroup$ Yes, in current English usage, "countable" does include "finite". $\endgroup$ – paul garrett Aug 16 '16 at 16:07
  • $\begingroup$ This is not an "if-then"-statement, at least not as it stands, so talking about contrapositive doesn't make that much sense. $\endgroup$ – Arthur Aug 16 '16 at 16:08
  • $\begingroup$ @paul I'm concerned about countably infinite thing. I'm afraid it'll make the given statement false. $\endgroup$ – Error 404 Aug 16 '16 at 16:11
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    $\begingroup$ @fleablood I was taught that countable means 'finite or countably infinite'. $\endgroup$ – Error 404 Aug 16 '16 at 16:18
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    $\begingroup$ The contrapositive of "Johny is absent on some days" is "It is not the case that Johny is present on all days". $\endgroup$ – fleablood Aug 16 '16 at 16:51

The statement $P(A)$ is a countable set for some $A$ is true, according to your definitions. In particular, it suffices to note that if $A =\{1\}$, then $P(A) = \{\emptyset,\{1\}\}$ is finite and therefore countable.

The negation of your statement is this: for all $A$, $P(A)$ is uncountable. This statement is false, since it's the negation of a true statement.

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    $\begingroup$ Oh! I confused negation with contrapositive. :( $\endgroup$ – Error 404 Aug 16 '16 at 16:33
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    $\begingroup$ The contrapositive of "P(A) is countable for some A" if "It is not true that P(A) is uncountable for all A". $\endgroup$ – fleablood Aug 16 '16 at 16:52
  • $\begingroup$ @fleablood thank you a million times!!! $\endgroup$ – Error 404 Aug 16 '16 at 17:10

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