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Universal and Existential Qualifiers

  1. Jan 26, 2016 #1
    1. The problem statement, all variables and given/known data
    Express the following statement using only quantifiers. (You may only use the set of Real and Natural Numbers)

    1. There is no largest irrational number.

    2. Relevant equations
    ##\forall=## for all
    ##\exists##=there exists

    3. The attempt at a solution
    I express the existence of irrational numbers by saying
    ##(\exists x \in \Re)(\forall p,q \in\mathbb{N})(\frac{p}{q}\neq x)##

    But now saying that x is not the largest irrational number is tricky to me. The book i am using said the answer would look quite complex.

    To prove that there is a bigger irrational number I begin by stating that another irrational number exists, and prove that is bigger.

    My thinking is that If I write:

    ##(\forall x \in \Re)(\exists y\in\Re)\wedge(\exists p,q,r,s \in \mathbb{N})\ni[{(\frac{p}{q}\neq x) \wedge(\frac{r}{s}\neq y)}\wedge( y>x)]##

    It proves that there is always a bigger irrational number than the one that is being considered, but i'm not completely sure my reasoning makes sense
  2. jcsd
  3. Jan 26, 2016 #2


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    The real number -1 is a problem. The natural number 0 is too.

    How would you say it if you can use other sets than ##\mathbb R## and ##\mathbb N##? I would start with that, and then try to rewrite the statement using only those sets.
  4. Jan 26, 2016 #3


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    Note that you are not proving this, but only trying to express the statement. You can also express statements that are false such as "there are no irrational numbers".
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