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Homework Help: Denumerable sets

  1. Dec 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove by induction that if set S = {x1,x2...,xn} is a set of n elements non of which are in a set A, then AuS is denumerable


    2. Relevant equations
    im not sure what to put here


    3. The attempt at a solution
    what does this mean
    denumerable? and how do i start this
    I know that i can say AuX when x is an element not in A
    but what do i do beyond this
     
  2. jcsd
  3. Dec 14, 2008 #2

    Dick

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    My definition is that a set is denumerable (or countable) if it's finite or can be put into 1-1 correspondence with the set of integers. What's your definition? I'm guessing A is given to be denumerable. Then if you already know AU{x} is denumerable it should be pretty easy. Think induction.
     
  4. Dec 14, 2008 #3

    HallsofIvy

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    If the problem asks you to prove a set is denumerable, you should already have been given a definition of "denumerable". I wouldn't be surprized if the definition were in the same chapter this problem is in. And, as tiny-tim suggested, you must have been given some information about A- without that the statement you say you want to prove is NOT true.
     
  5. Dec 14, 2008 #4
    ok i will look
    However, nothing more is said about a
    thats the whole problem
     
  6. Dec 14, 2008 #5

    Dick

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    If that's all that's said then you can't prove the set is denumerable. And if you haven't been given a definition of denumerable then you super can't prove it. And Halls, I'm not tiny-tim.
     
  7. Dec 14, 2008 #6
    ok so u are saying denumerable and countable are the same thing?
     
  8. Dec 15, 2008 #7

    Dick

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    That's what I would say, but check your book for fine print issues.
     
  9. Dec 15, 2008 #8

    HallsofIvy

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    Some books use "countable" to mean "there is a one to one function from the set onto N" and "denumerable" to mean finite or countable.

    If "Prove by induction that if set S = {x1,x2...,xn} is a set of n elements non of which are in a set A, then AuS is denumerable" is all that is said, then consider A= R. The statement is not true.
     
  10. Dec 15, 2008 #9

    HallsofIvy

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    But you look so much alike.:biggrin:

    (Seriously you and tiny tim have done wonderful work here- no wonder I get the two of you confused.)
     
  11. Dec 15, 2008 #10

    Dick

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    Well, since you put it so nicely, I guess I forgive you.
     
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