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Writing down an explicit bijection

  1. Aug 8, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]X= \{a,b,c\} [/itex] and [itex]Y= \{d,e\}[/itex]. Write down and explicit bijection [tex]N_{|X×Y|} → X×Y[/tex]

    3. The attempt at a solution
    Well I came up with the easiest method, just giving one value to each member of [itex]N_{|X×Y|}[/itex] so I was just wondering whether there is another way of doing it not by brute force? :)
     
  2. jcsd
  3. Aug 8, 2012 #2

    HallsofIvy

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    Is [itex]N_{|X\times Y|}[/itex] the set of positive integers from 1 to [itex]|X\times Y|[/itex]. Sounds like what you did is the simplest thing to do. [itex]X\times Y[/itex] contains 6 members so, write them in some order assign 1 to the first, 2 to the second, etc.
     
  4. Aug 8, 2012 #3
    One natural way to do this that generalizes to larger sets and Cartesian products with more than two factors is to use the lexicographic order. That's like alphabetical order using whatever order relations happen to be defined on the factors, going left to right in the Cartesian product.

    In the above case we'd have:

    (a, d)
    (a, e)
    (b, d)
    (b, e)
    (c, d)
    (c, e)
     
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