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Infinite set

  1. Apr 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that Square = {x ∈ N|∃y : N. x = y^2} is an infinite set.


    2. Relevant equations

    Definition 0.1. A set A is infinite if there is an injection f : A → A that is not also a surjection.

    3. The attempt at a solution

    I get how to show that there is an injection, but i dont get how to show that it is also not a surjection
     
  2. jcsd
  3. Apr 10, 2010 #2

    Office_Shredder

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    What injection are you looking at?
     
  4. Apr 10, 2010 #3
    to prove the injection i thought u could do something like this

    f(x) = f(y) for arbitrary x and y in N, then x^2 = y^2 and so x = y. and that proves an injection
     
  5. Apr 10, 2010 #4

    Mark44

    Staff: Mentor

    What is the above supposed to be saying?
     
  6. Apr 10, 2010 #5
    would it be that like 0 maps to 0, 1 to 1, 2 to 4, 3 to 9... so that shows injection but then like 5 doesnt map back to anything
     
  7. Apr 10, 2010 #6

    Office_Shredder

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    Except you're supposed to have a map that goes from the set of squares to the set of squares. But the function that you're looking at will still work
     
  8. Apr 10, 2010 #7
    so would it be like 4 mapped to 16 is an injection but if you go the other way 16 doesnt map back to 4 and that is why it isnt a surjection
     
  9. Apr 10, 2010 #8

    Mark44

    Staff: Mentor

    What set is this? {x ∈ N|∃y : N. x = y^2}

    Some characters are rendering as empty boxes.
     
  10. Apr 10, 2010 #9
    it is bascially the set of square numbers
     
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