Expressing the existence of irrational numbers

  • #1

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 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
 
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Answers and Replies

  • #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.
 
  • #3
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.
 
  • #4
WWGD
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No problem, sorry for the necessary nitpick.
 
  • #5
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?
 
  • #6
WWGD
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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.
 
  • #7
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.
 
  • #8
WWGD
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Glad it worked out.
 

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