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Math Help - student in concepts of math here.

  1. #1
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    student in concepts of math here.

    so I am an architecture student trying to get into the programming game so I'm taking a concepts of math corse at Carnegie Mellon. I'm horrible at writing proofs. I have no idea where to start even. but I get the concepts part. so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution. but how do I write the proof formally? am I even correct so far!? please help!
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  2. #2
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    Re: student in concepts of math here.

    Quote Originally Posted by ifoundfluffy View Post
    so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution.
    That's the right idea. To write a formal proof, you need to know definitions and auxiliary facts. When exactly is a set called countable? What theorems do you know about countability and cardinalities?
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  3. #3
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    Re: student in concepts of math here.

    I learned all about countable sets being plugged in to a function that outputs some N and is bijective. so I know how to do the proof if it were just R. I also know about cardinality and how for something to be bijective it must have the same cardinality.
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    Re: student in concepts of math here.

    Quote Originally Posted by ifoundfluffy View Post
    so I am an architecture student trying to get into the programming game so I'm taking a concepts of math corse at Carnegie Mellon. I'm horrible at writing proofs. I have no idea where to start even. but I get the concepts part. so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution. but how do I write the proof formally? am I even correct so far!? please help!
    If a set contains an uncountable subset than the set is uncountable.
    It is clear that \{(x,1):x\in\mathbb{R}\}\subset~\mathbb{R}\times \mathbb{N}.
    Map \{(x,1):x\in\mathbb{R}\}\to\mathbb{R} by \Phi: (x,1) \mapsto x.
    It follows that \Phi is a bijection so \{(x,1):x\in\mathbb{R}\} is uncountable, thus so is \mathbb{R}\times \mathbb{N}.
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  5. #5
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    Re: student in concepts of math here.

    Quote Originally Posted by ifoundfluffy View Post
    I learned all about countable sets being plugged in to a function that outputs some N and is bijective.
    Did your textbook or lecture notes give the definition using this language?

    I suggest you rigorously prove the fact that Plato used: If a set contains an uncountable subset than the set is uncountable.
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