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Math Help - reformulate "there are infinitely many primes"

  1. #1
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    reformulate "there are infinitely many primes"

    I was given this excersise as homework and to a certain extent i do understand what they are asking, but don't know where to start.

    Consider "there are infinitely many primes". Find a suitable universe U of elements so that this theorem can be reformulated in the form : A (is a subset of) B


    We were also told that we dont need to prove anything here.

    Thank you for any suggestions.
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  2. #2
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    Re: reformulate "there are infinitely many primes"

    I am not sure if this is the idea of the exercise, but if you formulate "there are infinitely many primes" as \forall n\in\mathbb{N}.\,P(n) for some property P, then it can also be stated as \mathbb{N}\subseteq\{n\in\mathbb{N}\mid P(n)\}.
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    Re: reformulate "there are infinitely many primes"

    Quote Originally Posted by Arturo_026 View Post
    I was given this excersise as homework and to a certain extent i do understand what they are asking, but don't know where to start.

    Consider "there are infinitely many primes". Find a suitable universe U of elements so that this theorem can be reformulated in the form : A (is a subset of) B


    We were also told that we dont need to prove anything here.

    Thank you for any suggestions.
    Would this homework be for a grade?
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  4. #4
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    Re: reformulate "there are infinitely many primes"

    Quote Originally Posted by Ackbeet View Post
    Would this homework be for a grade?
    Not directly. We're graded more on effort, but mainly on midterms and final.

    A simple set up I came up with (which was similar to his answer) was:

    Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

    Therefore A (is subset of) B (is subset of) U. ... sorry for not writting the symbol.

    Does this sound right?
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  5. #5
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    Re: reformulate "there are infinitely many primes"

    Quote Originally Posted by Arturo_026 View Post
    Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

    Therefore A (is subset of) B (is subset of) U. ... sorry for not writting the symbol.
    Why not learn to post in symbols? You can use LaTeX tags
    [tex]A\subseteq B [/tex] gives  A\subseteq B
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  6. #6
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    Re: reformulate "there are infinitely many primes"

    Quote Originally Posted by Arturo_026 View Post
    Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

    Therefore A (is subset of) B (is subset of) U.
    It is sufficient to take U to be the powerset of natural numbers. However, with A and B defined as above, A is infinite iff A\in B, not iff A\subseteq B.
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