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!

Please check my Contrapositive statement

  1. Feb 14, 2013 #1
    I am trying to write a CP for:

    Every connected M. Space with at least 2 points is uncountable.

    Restatement:

    if a MS X is connected with |X|≥ 2 => X is uncountable.

    Contrapositive:

    a MS X has only one point => X is not connected.

    Thanks
     
    Last edited: Feb 14, 2013
  2. jcsd
  3. Feb 14, 2013 #2
    I guess the correct statement is that:
    if X has only one point it is separated.
     
  4. Feb 14, 2013 #3
    How do I prove this though. the part about the singleton
     
  5. Feb 14, 2013 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If a space has only one point, it's clearly countable isn't it? Forming the CP isn't the challenge, it's showing a metric space with two points in it that's pathwise connected is uncountable. Any ideas on that one?
     
  6. Feb 15, 2013 #5

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You seemed to have changed "connected" into 'pathwise" connected. The OP was talking about just connected. The statement is true in both cases though (as I'm 100% sure you know), but it's much easier to prove if you take pathwise connected.
     
  7. Feb 15, 2013 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yep, true. I actually don't think it's hard for either form of connected. Thanks!
     
  8. Feb 15, 2013 #7

    pasmith

    User Avatar
    Homework Helper

    A metric space consisting of a single point is necessarily connected, is it not? The only non-empty open subset is the whole space.
     
  9. Feb 15, 2013 #8

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Given the statement "if a then b" (equivalently, ab), the contrapositive is "if not b then not a" (¬b → ¬a).

    Your right hand side in the first statement is "X is uncountable". The negation is "X is countable", which is how your contrapositive statement should start.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Please check my Contrapositive statement
  1. Check my work please? (Replies: 4)

  2. Please check my work (Replies: 2)

Loading...