# Need help with Countable Set

Discussion in 'Undergraduate Math' started by Herb, Feb 6, 2005.

1. ### HerbGuest

Q1.
Show that :
every infinite set has countable subset which is equipotent to N(atural numbers).

Could someone give me a hint or sketch of the proof ?

Q2.
Give a example :
Infinite set that is union of countable number of finite set.

My answer : \/_{n=1 to infinity} {n}

Is it correct ?

Herb, Feb 6, 2005

2. ### William ElliotGuest

As the set S is infinite, there is a injection f:S -> S
with f(S) proper subset S. Pick x in S\f(S).

Show { f^k(x) | k in N } is countable.
Yes, it'll do.
What's incorrect is to write hard to read equations like
ax^2+bx+c=(x-r)(x-s)=k+5
I rarely bother with posts like that for there are so many more
that are easier to read, they usually occupy my evening's time.

William Elliot, Feb 6, 2005

3. ### HerbGuest

Thank you. Sir
Yes, I believe this, too.
But, I cannot think method for proving this.
How can I prove above fact ?

Herb, Feb 8, 2005
4. ### William ElliotGuest

It's Cantor's definition of infinite.
What definition of infinite are you using?

William Elliot, Feb 8, 2005