• Support PF! Buy your school textbooks, materials and every day products Here!

Countable subet

So, the task is to prove: Every infinite set has a infinite countable subset.


2. A set [tex]S[/tex] is countable if there exists a bijection [tex]\phi: \mathbb{N}\rightarrow X[/tex]


3. The Attempt at a Solution
 

Answers and Replies

You can easily construct a countable subset [itex]\{s_1,s_2,\dots\}[/itex] wiht [itex]s_1,s_2,\dots[/itex] being elements of S.

Let [itex]s_1[/itex] be some element of S. Let inductively [itex]s_n[/itex] be some element of [itex]S-\{s_1,\dots,s_{n-1}\}[/itex].

Can you show that this works for any infinte set, and that it does not work for any finite set?
 
Last edited:

Related Threads for: Countable subet

  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
1K
Replies
3
Views
2K
Replies
6
Views
5K
  • Last Post
Replies
3
Views
746
  • Last Post
Replies
1
Views
509
  • Last Post
Replies
3
Views
982
  • Last Post
Replies
4
Views
5K
Top