1. Not finding help here? Sign up for a free 30min 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!

Bijection proof for set products

  1. Oct 13, 2011 #1
    Let A1, A2, T be non-empty sets such that A1 is bijective to A2.

    Show that A1 × T is bijective to A2 × T



    So far I've been able to show that for any b where b is an element of A2, there must be some a within A1 such that f(a) = b. Ive been able to do the same for proving injectivity between A1 and A2. I just cant figure out how to apply this to A1 x T and A2 x T.
     
  2. jcsd
  3. Oct 13, 2011 #2
    Let [itex]f: A_1 \rightarrow A_2[/itex] be a bijection between [itex]A_1[/itex] and [itex]A_2[/itex]. Can you find a bijection between [itex]A_1 \times \ T[/itex] and [itex]A_2 \times \ T[/itex]?
     
  4. Oct 13, 2011 #3
    exactly, Im pretty sure I need to prove that for some function g: A1 x T --> A2 x T , any (a2, t2) that is an element of A2 x T must have some element (a1,t1) in A1 x T such that g(a1,t1) = (a2,t2). Im just not sure how to show this. I have the same problem with showing the injective aspect of the bijection proof.
     
  5. Oct 14, 2011 #4
    Here's another hint: If T is any non-empty set, you can always find a bijection between T and itself.
     
  6. Oct 14, 2011 #5

    Deveno

    User Avatar
    Science Advisor

    given the bijection f, is k:A1 x T → A2 x T

    given by k(a1,t) = (f(a1),t) a bijection?
     
  7. Oct 14, 2011 #6
    but is it necessarily true that T must be a bijection to itself?
     
  8. Oct 14, 2011 #7

    Deveno

    User Avatar
    Science Advisor

    isn't the function t→t, for all t in T a bijection? let's give it a name, we'll call it g.

    so g(t) = t, for all t in T.

    is it unclear to you whether or not this is a bijection?
     
  9. Oct 14, 2011 #8
    I see how that specific function is a bijection. But I dont see how all functions from T to T would have to be bijections.
     
  10. Oct 14, 2011 #9

    Deveno

    User Avatar
    Science Advisor

    we don't need to find "all" bijections between A1 xT and A2 x T. just one will do.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Bijection proof for set products
  1. Bijection Proof (Replies: 15)

Loading...