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Math Help - Trouble with Quick Proof

  1. #1
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    Trouble with Quick Proof

    Let A and B be sets.

    Prove that if A is infinite and A C B, then B is infinite.

    My guess would be:

    A is infinite so A=/= the empty set and A is not equivalent to Nk for some natural number k.

    So obviously I know that since B is bigger than A, then B is clearly infinite, but I really don't know how to show this.
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  2. #2
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    Quote Originally Posted by zhupolongjoe View Post
    Let A and B be sets.
    Prove that if A is infinite and A C B, then B is infinite.
    My guess would be:
    A is infinite so A=/= the empty set and A is not equivalent to Nk for some natural number k.
    It appears that your text material use the definition, “A is infinite if A is not finite”.
    If B were finite, what does that imply about A?
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  3. #3
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    That is indeed the definition in the text.

    If B were finite, then A would be finite

    I suppose it should be a proof by contrapositive?

    I.e. Assume B is finite.

    Then since A C B, A is finite, and so the contrapositive is true.

    Is it that simple?
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  4. #4
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    Quote Originally Posted by zhupolongjoe View Post
    Assume B is finite.
    Then since A C B, A is finite, and so the contrapositive is true.
    Is it that simple?
    Indeed it is.
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