High School What is the converse statement of the given sentence?

[Danijel]

The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement? Would it be:"If there exists a natural number n bigger than some number x, then every single x is real"?

 

[fresh_42]

The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement? Would it be:"If there exists a natural number n bigger than some number x, then every single x is real"?

The "if" that you inserted isn't necessary and complicates it. In general you transform each $\forall \longleftrightarrow \exists$ and negate the statement. Thus we get $(\exists x \in \mathbb{R}) (\forall n \in \mathbb{N})\, : \, n \leq x\,$.

 

[Danijel]

The "if" that you inserted isn't necessary and complicates it. In general you transform each $\forall \longleftrightarrow \exists$ and negate the statement. Thus we get $(\exists x \in \mathbb{R}) (\forall n \in \mathbb{N})\, : \, n \leq x\,$.

But isn't that exactly the negation of the statement, not the converse? I am interested in "if Q, then P" if the given statement is "If P, then Q", even though this statement isn't in the if - then form. Sorry if I didn't understand you.

 

[fresh_42]

First sorry, for not understanding you. Yes, it was the negation. The statement says: If $x$ is a real number, then we can find a natural number $n > x$. The conversion is a bit strange: If we can find a natural number $n$ which is greater than a given number $x$, then $x$ is real. This is because $\in \mathbb{R}$ is the only statement on the left. But a nice example on how to get a completely different statement by simply turning the direction of conclusion. A method which is often used by politicians.

 

[PeroK]

I would take the converse to be the second part of an iff.

Here you have: If x has property P, then x has property Q."

The converse of this is: If x has property Q, then x has property P.

 

[Danijel]

I would take the converse to be the second part of an iff.

Here you have: If x has property P, then x has property Q."

The converse of this is: If x has property Q, then x has property P.

Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.

 

[fresh_42]

Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.

This is due to the fact, that we don't have a real conclusion. We have a statement here, that a certain set is not empty.

 

[PeroK]

Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.

If it's Friday, X goes to the big cinema.

Converse:

If X goes to the big cinema, then it's Friday.

The converse would not involve changing the "big" cinema to the "small" cinema here.

 

[Danijel]

If it's Friday, X goes to the big cinema.

Converse:

If X goes to the big cinema, then it's Friday.

The converse would not involve changing the "big" cinema to the "small" cinema here.

If it's Friday, X goes to the big cinema.

Converse:

If X goes to the big cinema, then it's Friday.

The converse would not involve changing the "big" cinema to the "small" cinema here.

Right, so "exists" stays with the n and "all" stays with x, they don't swap places? Thank you anyway.

 

[PeroK]

Right, so "exists" stays with the n and "all" stays with x, they don't swap places? Thank you anyway.

The converse of: "All logicians are nerds" is "All nerds are logicians.".

 

[mfb]

The sentence is : "For all real numbers there exists a natural number that is smaller".

Is that supposed to be "smaller"? That is (a) wrong and (b) not what the rest of the thread uses.

 

[Danijel]

Is that supposed to be "smaller"? That is (a) wrong and (b) not what the rest of the thread uses.

What I meant by smaller is n<x. English is not my mother tongue and I am probably not using words the way they are supposed to be used.

 

[PeroK]

What I meant by smaller is n<x. English is not my mother tongue and I am probably not using words the way they are supposed to be used.

From your posts on this thread it is difficult to find anything that gives you away as a non-native speaker.

 

[Danijel]

From your posts on this thread it is difficult to find anything that gives you away as a non-native speaker.

Not sure if a compliment or a reference that I do not really understand the subject.

 

[PeroK]

Not sure if a compliment or a reference that I do not really understand the subject.

It was meant as a compliment.

 

[Danijel]

It was meant as a compliment.

Thank you. :)

 

[mfb]

What I meant by smaller is n<x

Well, you used n>x in your post, that lead to my question.

The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement?

 

[Danijel]

Well, you used n>x in your post, that lead to my question.

You are right, I said smaller, but meant greater. Sorry.