Expressing the existence of irrational numbers

In summary, the conversation discussed the use of existential and universal quantifiers to express the existence of rational and irrational numbers in sets of real and natural numbers. The correct way to express the existence of irrational numbers is through the negation of the rational numbers, using the predicate of equality.
  • #1
TyroneTheDino
46
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 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
 
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  • #2
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
WWGD said:
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
No problem, sorry for the necessary nitpick.
 
  • #5
WWGD said:
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
TyroneTheDino said:
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
WWGD said:
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
Glad it worked out.
 

1. What is an irrational number?

An irrational number is a number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a simple fraction, and its decimal representation does not terminate or repeat.

2. How do we know that irrational numbers exist?

We know that irrational numbers exist because they have been proven mathematically through various proofs, such as the proof that √2 is irrational. Additionally, irrational numbers are necessary to fill in the gaps between rational numbers on the number line.

3. Can irrational numbers be written in any other form?

Irrational numbers cannot be written as fractions, but they can be written in other forms such as decimal expansions or as infinite series. However, these forms never fully capture the exact value of an irrational number, as it has an infinite number of digits after the decimal point.

4. Are all irrational numbers transcendental?

No, not all irrational numbers are transcendental. Transcendental numbers are a subset of irrational numbers that cannot be the root of a polynomial equation with rational coefficients. Examples of transcendental numbers include π and e.

5. How do irrational numbers impact our understanding of mathematics?

Irrational numbers play a crucial role in mathematics, as they allow us to solve equations and problems that would otherwise be impossible to solve. They also help us understand the concept of infinity and the infinite nature of the universe.

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