Universal and Existential Qualifiers

[TyroneTheDino]

Homework Statement

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.

Homework Equations

$\forall=$ for all
$\exists$=there exists

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

 

[Fredrik]

The Attempt at a Solution

I express the existence of irrational numbers by saying
$(\exists x \in \Re)(\forall p,q \in\mathbb{N})(\frac{p}{q}\neq x)$

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

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.

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.

 

[PeroK]

Homework Statement

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.

Homework Equations

$\forall=$ for all
$\exists$=there exists

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

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".