# How many left?

• Aug 17th 2010, 08:35 AM
devami
How many left?
We have an infinite number of apples. Spread them on an infinite set of identical number of groups.In each group of 5 apples.Now from each group to take out 2 apples. It remains to 3.
The remainder of the set split into an infinite set of groups.In each group, and 7 apples. Now take out for 2 apples. It remains for 5 apples.
The remainder of the set split into an infinite set of groups.In each group, and 11 apples. Now take out for 2 apples. It remains for 9 apples.

And so on indefinitely. Seizure 2 / n. Save n-2 / n. N-primes in order.Question: how many apples will be left? Finite or infinite?
• Aug 24th 2010, 07:45 PM
tonio
Quote:

Originally Posted by devami
We have an infinite number of apples. Spread them on an infinite set of identical number of groups.In each group of 5 apples.Now from each group to take out 2 apples. It remains to 3.
The remainder of the set split into an infinite set of groups.In each group, and 7 apples. Now take out for 2 apples. It remains for 5 apples.
The remainder of the set split into an infinite set of groups.In each group, and 11 apples. Now take out for 2 apples. It remains for 9 apples.

And so on indefinitely. Seizure 2 / n. Save n-2 / n. N-primes in order.Question: how many apples will be left? Finite or infinite?

I'm afraid the question is loosely posed, to say the least: if you have an infinite number of apples (countable infinite? A continuum...?), and you "Spread them on an infinite set of identical number of groups"...what does this mean?

Now, no matter what kind of infinite number of apples you have, if you "divide" them in groups of 5 then the cardinality of this set of groups will have to be the same as the cardinality of the set of apples, and if you now take two out of each of this groups then you're left again with exactly the same cardinality of apples as in the beginning, and YOU are implicitly assuming this since in the 2nd stage of your question you once again carry on the same procedure as above, but with 7 apples instead of five, which of course it's just a minor change.

So each time you're left with the same cardinality of apples as in the beginning...

Tonio