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Math Help - Proving that AUB is countable Infinite given a specific case

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    Proving that AUB is countable Infinite given a specific case

    (a) Prove: If A is a finite set and B is a countable infinite set and A (intersection) B = null set, then AUB is
    countable infinite.
    (b) Is 'A (intersection) B = null set ' a necessary assumption?


    For A, I understand that I need to find a bijection f:B to Natural numbers, then prove that it's 1-1 and onto by proving its invertiblity.

    However, where I get confused is how to incorporate that A is a finite set. After proving that B is countable infinite and that it has a bijection with N, then is there a way to prove that if A is finite, then AUB is also countably infinite?

    Thank you.
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    Re: Proving that AUB is countable Infinite given a specific case

    Quote Originally Posted by nexttime35 View Post
    (a) Prove: If A is a finite set and B is a countable infinite set and A (intersection) B = null set, then AUB is countable infinite.
    (b) Is 'A (intersection) B = null set ' a necessary assumption?
    I am sure that you understand that any proof depends upon the definitions in use.
    The definitions of finite do differ.

    But in any case, if A is finite the \exists k\in\mathbb{N} such that A\sim\{0,\cdots,k\}

    If A\cap B=\emptyset them just shift the map by k.

    If A\cap B\ne\emptyset them just shift the map by k-|A\cap B|.
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