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Expressing the existence of irrational numbers

  1. Jan 28, 2016 #1
    1. The problem statement, all variables and given/known data
    Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers

    2. Relevant equations


    3. The attempt at a solution
    I believe the existance of rational numbers can be stated as:

    ##(\forall n \in \Re)(\exists p,q \in \mathbb{N}) \ni [(p \mid q)=x]##

    So to say that there are irrational numbers is the negation of this being:

    ##\neg (\forall x \in \Re)(\exists p,q \in \mathbb{N}) \ni [(p \mid q)=x]##

    Which becomes

    ##(\exists x\in \Re)\ni(\forall p, q \in \mathbb{N})[(p\nmid q )\ne x]##

    Is my rationality correct
     
    Last edited: Jan 28, 2016
  2. jcsd
  3. Jan 28, 2016 #2

    WWGD

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    If I understood correctly, the right-most expression should be ##p|q \neq x ##. Just slide the negation parenthesis by parenthesis according to usual negation rules.
     
  4. Jan 28, 2016 #3
    Oh thank you, correction made.
     
  5. Jan 28, 2016 #4

    WWGD

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    No problem, sorry for the necessary nitpick.
     
  6. Jan 29, 2016 #5
    But one question:

    since i had ##\neg[(p\mid q)=x]##
    when the negation moves inside the expression it becomes:
    ##[(p\nmid q )\ne x]##
    Correct?
     
  7. Jan 29, 2016 #6

    WWGD

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    Well, no, because the predicate , or the content of the original statement is that the expression p/q =x . The statement is not about whether p divides q.
    The predicate of the original is that there is an equality between the object p|q and the object x. Here p|q is sort of the subject of the sentence, not part of the content of what is being asserted; the content is that the expression p|q equals x. Maybe you can make this more rigorous by using a predicate of something like ##R_{pqx} ##. More formally, ##p|q## is a member of your universe of discourse, i.e., it is an element of
    the collection of objects about which you are stating something. p|q is 'the subject' and = is the predicate, i.e., what is being stated about p|q. You negate the predicate/relation , not the subject.
     
  8. Jan 29, 2016 #7
    Ah I understand, this makes more sense to me now. Thank you.
     
  9. Jan 29, 2016 #8

    WWGD

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    Glad it worked out.
     
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