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Homework Help: Set theory and ordered pairs

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Determine whether or not the following definition of an ordered pair is set theoretic (i.e. you can distinguish between the "first" element and the "second" element using only set theory).
    (x,y)={x,{y}}

    2. Relevant equations



    3. The attempt at a solution
    I am inclined to think no, but can anyone explain the rationale for why it does not work?
     
    Last edited: Sep 26, 2010
  2. jcsd
  3. Sep 26, 2010 #2

    Dick

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    x is an element of {x,{y}}. Is y an element of {x,{y}}?
     
  4. Sep 26, 2010 #3
    No. But why does it matter if it is in it or not?
     
  5. Sep 27, 2010 #4

    Dick

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    You asked whether you can distinguish between x and y using set theory. Being 'an element of a set' is set theory, isn't it?
     
  6. Sep 27, 2010 #5

    Hurkyl

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    Here is an ordered pair, by this definition.
    {{2}, {1}}​

    Can you tell me what the left element is, and what the right element is?
     
  7. Sep 27, 2010 #6

    Dick

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    The definition of (x,y) is {x,{y}}, isn't it? Not {{x},{y}}.
     
  8. Sep 27, 2010 #7

    Hurkyl

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    Remember that we're doing set theory, so x and y are variables denoting sets. There exists x and y such that (x,y)={{2},{1}}.

    (Also, my 1 and 2 are meant to be naming sets whose precise identity isn't relevant. The typical naming scheme here is 0={}, 1={0}, 2={0,1}, .... Mainly, I just didn't want to have millions of braces lying around in my example)
     
  9. Sep 27, 2010 #8

    Dick

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    Ok, I see what you are saying. {{2},{1}} corresponds to either the ordered pair ({2},1) or ({1},2). If I know x and y, I can tell you which is first. But if I don't know what x and y are to begin with, I can't tell you what they are from {{2},{1}}. Thanks.
     
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