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 said: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"?
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 said: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\,##.
Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.PeroK said: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.
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.Danijel said:Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.
Danijel said:Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.
PeroK said: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 said: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.
The converse of: "All logicians are nerds" is "All nerds are logicians.".Danijel said:Right, so "exists" stays with the n and "all" stays with x, they don't swap places? Thank you anyway.
Is that supposed to be "smaller"? That is (a) wrong and (b) not what the rest of the thread uses.Danijel said:The sentence is : "For all real numbers there exists a natural number that is smaller".
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.mfb said:Is that supposed to be "smaller"? That is (a) wrong and (b) not what the rest of the thread uses.
Danijel said: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.
Not sure if a compliment or a reference that I do not really understand the subject.PeroK said:From your posts on this thread it is difficult to find anything that gives you away as a non-native speaker.
It was meant as a compliment.Danijel said:Not sure if a compliment or a reference that I do not really understand the subject.
Thank you. :)PeroK said:It was meant as a compliment.
Well, you used n>x in your post, that lead to my question.Danijel said:What I meant by smaller is n<x
Danijel said: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?
You are right, I said smaller, but meant greater. Sorry.mfb said:Well, you used n>x in your post, that lead to my question.
The converse statement of a given sentence is formed by switching the hypothesis and the conclusion of the original statement. This means that if the original statement is "If A, then B", the converse statement would be "If B, then A".
The converse statement is a way of expressing the same logical relationship between two statements, but in reverse order. It is often used to test the validity of a statement, as a true statement will have a true converse statement.
No, the converse statement can only be true if the original statement is true. If the original statement is false, it does not necessarily mean that the converse statement is false, but it cannot be proven to be true.
In mathematical proofs, the converse statement can be used to prove the validity of a statement. If the converse statement is also true, it can be used as evidence to support the original statement. However, the converse statement alone cannot be used as proof of the original statement.
Yes, there are some statements where the converse statement is not equivalent to the original statement. These are known as "biconditional" statements, where the converse statement is also true. An example of this is "A square has four equal sides" and "A figure with four equal sides is a square". In this case, both the original statement and the converse statement are true.