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Mathematical Reasoning and Writing - Counterexamples with subsets.

  1. Apr 1, 2013 #1
    1. The problem statement, all variables and given/known data

    Let f: A --> B be a function and let S, T [itex]\subseteq[/itex] A and U, V [itex]\subseteq[/itex] B.

    Give a counterexample to the statement: If f (S) [itex]\subseteq[/itex] f (T); then S [itex]\subseteq[/itex] T:

    2. Relevant equations



    3. The attempt at a solution

    PF:

    Assume f(S) [itex]\subseteq[/itex] f(T).

    Let x [itex]\in[/itex] S.

    Then [itex]\exists[/itex] y [itex]\in[/itex] f(S) [itex]\ni[/itex] f(x) = y.

    Since f(S) [itex]\subseteq[/itex] f(T), y [itex]\in[/itex] f(T).

    ****

    Suppose [itex]\forall[/itex] a [itex]\in[/itex] T where a ≠ x, [itex]\exists[/itex] y [itex]\in[/itex] f(T) [itex]\ni[/itex] f(a) = y.

    Then x [itex]\notin[/itex] T.

    Q.E.D.



    I am not exactly sure I am doing this right, especially the reasoning beyond the ****. I almost have the feeling I should use the pre image of f(T) somehow to show that x [itex]\notin[/itex] T.

    Why can I not just say that [itex]\exists[/itex] x [itex]\in[/itex] S where x [itex]\notin[/itex] T? Would that not suffice as a counterexample in such a general proof as this?
     
  2. jcsd
  3. Apr 1, 2013 #2

    Dick

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    You aren't supposed to do a general proof. The statement isn't false for all functions, only some. You have to think of one.
     
  4. Apr 1, 2013 #3
    Oh! I don't know why I was thinking what I was.

    How about...

    Proof:
    Let f: ℝ → [0, ∞) by f(x) = x2.
    Q.E.D.

    The negative ℝ could be considered S and the positive ℝ could be considered T. Then by f(x) = x2, f(S) is contained in f(T), but obviously S is not contained in T.

    Would this be a suitable counterexample?
     
  5. Apr 1, 2013 #4

    micromass

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    That's perfect!
     
  6. Apr 1, 2013 #5

    micromass

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    Some comments on your OP. When writing a proof, you should never use the symbols ##\exists## and ##\forall##. You should always write it out in words. This is a very common mistake that new people make and it's one way I see whether somebody is used to proving things or not.

    And your proof should be much wordier. A proof should really be read like an english text.
     
    Last edited: Apr 1, 2013
  7. Apr 1, 2013 #6

    Curious3141

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    I had no idea walruses observe the 1st of April. :biggrin:
     
  8. Apr 1, 2013 #7

    micromass

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    I was serious :frown:
     
  9. Apr 1, 2013 #8

    Curious3141

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    Haha, really? :biggrin:

    Seriously, I do find a proof with all those logical operators a real PITA to read. I'd much prefer they wrote it all in words. But then, I'm not a mathematician, your walrus-ness. :tongue2:
     
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