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'.