Expressing the existence of irrational numbers

[TyroneTheDino]

Homework Statement

Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers

Homework Equations

The Attempt at a Solution

I believe the existence 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

 

[WWGD]

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.

 

[TyroneTheDino]

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.

Oh thank you, correction made.

 

[WWGD]

No problem, sorry for the necessary nitpick.

 

[TyroneTheDino]

No problem, sorry for the necessary nitpick.

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?

 

[WWGD]

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?

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.

 

[TyroneTheDino]

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.

Ah I understand, this makes more sense to me now. Thank you.

 

[WWGD]

Glad it worked out.