1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cardinality of infinite sets

  1. Nov 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that if A,B, and C are nonempty sets such that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|, then |A|=|B|

    3. The attempt at a solution
    Assume B [itex]\subset[/itex] C and A [itex]\subset[/itex] B (else A=B or B=C), and there must be a bijection f:A[itex]\rightarrow[/itex]C...
     
  2. jcsd
  3. Nov 27, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    So far you are just stating what the problem told you. Don't you have some theorems you might apply?
     
  4. Nov 27, 2012 #3

    Mark44

    Staff: Mentor

    Why not start with the given condition, that A [itex]\subseteq[/itex] B [itex]\subseteq[/itex] C and |A|=|C|?
     
  5. Nov 27, 2012 #4
    I know, I don't really know where to start. Schroder-Bernstein maybe?
     
  6. Nov 27, 2012 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's the one! Try and apply it. Here's a hint. If A is subset of B, then there's an injection from A into B, right?
     
    Last edited: Nov 27, 2012
  7. Nov 27, 2012 #6
    Right. And we want to show there's an injection from B to A. Would it be that since there's an injection from B to C and and injection from C to A, there must be an injection from B to A?
     
  8. Nov 27, 2012 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure. That's wasn't so hard, was it? You might want to spell out some of the details, like what the actual injections are in terms of your bijection f:A->C. But that's the idea.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cardinality of infinite sets
  1. Cardinality of set (Replies: 4)

Loading...