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Math Help - Cardinal numbers proof

  1. #1
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    Cardinal numbers proof

    How can I prove these lines:

    Let N be a set of natural numbers and |N|=Xo.
    How can i prove that
    Xo + k = Xo
    or
    Xo * Xo = Xo

    Help anyone?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tashe View Post
    How can I prove these lines:

    Let N be a set of natural numbers and |N|=Xo.
    How can i prove that
    Xo + k = Xo
    i assume k is a cardinal number such that k \le \aleph _0. otherwise, it is not true

    we can prove the claim by showing:

    (1) \aleph _0 + k \ge \aleph _0 and (2) \aleph _0 + k \le \aleph _0

    (1) is immediate.

    for (2): \aleph _0 + k \le \aleph _0 + \aleph _0 = 2 \aleph _0 = \aleph _0 ....there should be a theorem in your book telling you something to that effect

    or
    Xo * Xo = Xo

    Help anyone?
    hint: \aleph _0 \cdot \aleph _0 is the cardinal number of \mathbb{N} \times \mathbb{N} by definition. you can prove that this cardinal number is also infinitely countable, which will show that it is equal to \aleph _0. the way i usually see this done is by the use of a diagram, you should also see a proof using this technique in your text. the diagram looks like a grid, with the elements of \mathbb{N} along the first row and column, and the elements in the inner cells are there corresponding products. you count off the elements in diagonals ...
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