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Some Simple Questions (Invertibility, Relations)

  1. Mar 27, 2008 #1
    1. The problem statement, all variables and given/known data

    A.) Jon wants to define a function f: A->B as invertible iff for all a in A and all b in B with f(a)=b, there exists a function g:B->A for which g(b)=a.

    Is that reasonable?

    B.) Determine Whether the relation ~ on the Real Numbers defined by x~y is reflexive, symmetric, or transitive.

    1.) x~y iff xy<= 0
    2.) x~y iff xy < 0

    2. Relevant equations

    None really, except maybe a definition for invertible.

    3. The attempt at a solution
    this seems to make sense, but it seems odd to answer a math question with a "yes" and move on. Am I missing something about the defininition of invertibility that makes the statement in the question incorrect?

    For B, these questions seem really easy, but they also seem to be exactly the same. Both relations seem to be Symmetric only....because x^2 is not less than zero for all real values, and the counterexample x=-1, y=1, z=-1 proves that both aren't transitive. AM i missing something?
  2. jcsd
  3. Mar 27, 2008 #2


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    Science Advisor
    Homework Helper

    A) looks a little subtle. Think about it. For one thing the quantifiers smell wrong. You said for all a and b there exists a function g. Jon didn't say that the g should be the same for ALL choices of a and b. Second, worry about the case where f isn't onto (surjective). What is your definition of 'invertible'? B) looks pretty reasonable to me.
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