# Universal and Existential Qualifiers

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1. Jan 26, 2016

### TyroneTheDino

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. Jan 26, 2016

### Fredrik

Staff Emeritus
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.

3. Jan 26, 2016

### PeroK

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".