Where does 0 fit in?
True or False?
Let S be the set that contains all the integers x such that x x is a positive integer.
And let S ′ be the set that contains all the integers x such that x x is not a positive integer.
Then S and S ′ have equal cardinalities .
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1 solution
S' will be an empty set because anything raised to a negative power will not give an integer except -1 and it is clearly mentioned that S' is a set of integers. I'm confused!!
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S ′ is anything that [produces something] is not a positive integer. That includes pink elephants.
Bracket comment [...] later added for clarity.
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:)... Then how come their cardinality is equal ?So you mean the question is wrong! In the question it is saying that S' contains integers x^x except positive integers but any negative integer raised to a negative power will not give an integer.
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@Nashita Rahman – Okay, I think I know where the confusion lies. The set S ′ is the set of all integers < 1 , which produces non-integer results from the expression x x . Neither set S nor S ′ lists the results from x x , they only list the integers that go into that function x x . So, set S is all the integers ≥ 1 , and set S ′ lists all the integers < 1 , so that the cardinality of both S and S ′ are the same.
"...saying that S' contains integers x^x...." is where you are already getting derailed. It should have read "...saying that S' contains integers x such that x^x [ produces something other than positive integers ] ".
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@Michael Mendrin – Okay , now I got it! I really did a very silly mistake.Thank You for clearing the confusion.
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@Nashita Rahman – I should have been more clear. See now my first reply to you, and the note about added bracket comment.
For x = all natural numbers 1 , 2 , 3 , 4 , . . . ) , S is a positive integer.
For x ′ = all other integers ( 0 , − 1 , − 2 , − 3 , . . . . ) , S ′ is not a positive integer, with the special case of x = 0 where S ′ is indeterminate.
A simple bijection is possible, x ′ ↔ ( 1 + x ) , so S and S ′ have the same cardinalities.