Thread: prove that A contains a subset of size n

1. prove that A contains a subset of size n

If A is an infinite set, prove that A contains a subset of size n for any n>=1.

anyone can provide some hints for starting the proof?

2. Re: prove that A contains a subset of size n

Induction seems like it will be useful here, and I believe you'll need to invoke the axiom of choice during the induction step to choose an element of A which is not in your subset of size n

3. Re: prove that A contains a subset of size n

what is the definition of the axiom of choice ?

4. Re: prove that A contains a subset of size n

Originally Posted by wopashui
If A is an infinite set, prove that A contains a subset of size n for any n>=1.
anyone can provide some hints for starting the proof?
@wopashui, I have meant to post this bit of advice on your other questions about infinite sets. It may surprise you, but there are several different ways to define infinite sets.
Some authors say a set infinite if it is not finite.
Others say a set infinite if it is equipotent to a proper subset of itself.
Still others require an infinite set to contain a copy of the natural numbers.

While any of these lead to the same set of theorems and problems, the definition you are given it makes an enormous difference in the way one writes up a solution.

So how does your textbook/instructor define infinite set?

5. Re: prove that A contains a subset of size n

hi Plato, the definition of infinite set we use is" the set that is not finite. but we also proved that any infinite set is equivalent to a proper subset of itself