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Math Help - Proofs

  1. #1
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    Proofs

    Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
    1. To prove that every subset from a countable set is countable
    2. To prove the following senstences:
    a. A is denumerable
    b. Exists the function f: A--> N injective
    c. Exists the function f :N--> A onto
    Thanks Friends
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  2. #2
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    Quote Originally Posted by user View Post
    Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
    1. To prove that every subset from a countable set is countable
    2. To prove the following senstences:
    a. A is denumerable
    b. Exists the function f: A--> N injective
    c. Exists the function f :N--> A onto
    Thanks Friends
    How about your telling us what about these you don't understand?
    How does your text define finite and denumerable?

    Can you show some work?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by user View Post
    Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
    1. To prove that every subset from a countable set is countable
    2. To prove the following senstences:
    a. A is denumerable
    b. Exists the function f: A--> N injective
    c. Exists the function f :N--> A onto
    Thanks Friends
    Quote Originally Posted by Plato View Post
    How about your telling us what about these you don't understand?
    How does your text define finite and denumerable?

    Can you show some work?
    I agree with Plato, some work would be nice and the terms finite and deumerable are a tad sketchy.

    For though notice that for some countable set \mathcal{T} there exists some mapping f:\mathcal{T}\mapsto\mathbb{N} such that f is bijective. Thus for some subset \mathcal{I}\subseteq\mathcal{T} clearly the restriction f_{\mathcal{I}}:\mathcal{I}\mapsto\mathbb{N} given by f_{\mathcal{I}}(n)=f(n)\quad\forall n\in\mathbb{I} is an injection....soo
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    IFor though notice that for some countable set \mathcal{T} there exists some mapping f:\mathcal{T}\mapsto\mathbb{N} such that f is bijective.
    I am sure that some author somewhere has done that.
    However, most often f:\mathcal{T}\mapsto\mathbb{N} such that f is bijective means the set \mathcal{T} is deumerable.
    Countable sets are finite or deumerable (countablely infinite).
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Plato View Post
    I am sure that some author somewhere has done that.
    However, most often f:\mathcal{T}\mapsto\mathbb{N} such that f is bijective means the set \mathcal{T} is deumerable.
    Countable sets are finite or deumerable (countablely infinite).
    That's true. In that case, replace bijection with injection.
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