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Definition of Ordered Pair

  1. Nov 25, 2009 #1
    The Kuratowski definition of the ordered pair is (a,b) = {{a},{a,b}}...this sort of lost me...how did they define an ordered pair (in which order of elements matters) using set notation (how does this definition work)?
     
  2. jcsd
  3. Nov 25, 2009 #2
    An ordered pair (a,b) is supposed to be some object with this property:

    (a,b) = (c,d) if and only if a=c and b=d.

    What it actually *is* we don't care, as long as it has this property. Kuratowski wrote down his definition as something that has this property.
     
  4. Nov 25, 2009 #3
    I see...I thought they were defining (a,b) with the expression. So, in that sense, the expression does not show that order matters, right?
     
  5. Nov 25, 2009 #4
    Yes it does. Notice that if a is not equal to b, the RHS becomes {{b}, {b,a}} which is definitely not equal to {{a}, {a,b}}. Also, if {{a}, {a,b}} = {{x}, {x,y}}, then necessarily a = x and b = y.

    This is just a way of capturing the notion of ordered pairs with a set-theoretic definition. What ordered pairs actually are, like g_edgar said, doesn't really matter: as long your notion/definition of them has the desired properties, then there is no harm done.
     
  6. Nov 25, 2009 #5
    I see...Thanks for the replies...but isn't {{a},{a,b}} just another way of writing {a,b}?
     
  7. Nov 26, 2009 #6
    No... {a,b} is not the same as {{a}, {a,b}}. The members of the first set are a and b, the members of the second are {a} and {a,b}. The members of the second set are NOT a and b, if this is what you are implying.
     
  8. Nov 26, 2009 #7

    HallsofIvy

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    {1,3} is the same as {3,1}. But {{1},{1,3}} is not the same as {{3},{3,1}}. Kuratowski's definition basically states that there are two elements and distinguishes between the two, thus giving an order.
     
  9. Nov 28, 2009 #8
    Thus, the elements are different...I think I see whats going on to a better extent. Thanks for the replies.
     
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