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B What is the converse statement of the given sentence?

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  1. Dec 13, 2017 #1
    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"?
     
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  3. Dec 13, 2017 #2

    fresh_42

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    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\,##.
     
  4. Dec 13, 2017 #3
    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.
     
  5. Dec 13, 2017 #4

    fresh_42

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    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.
     
  6. Dec 13, 2017 #5

    PeroK

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    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.
     
  7. Dec 13, 2017 #6
    Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.
     
  8. Dec 13, 2017 #7

    fresh_42

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    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.
     
  9. Dec 13, 2017 #8

    PeroK

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    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.
     
  10. Dec 13, 2017 #9
    Right, so "exists" stays with the n and "all" stays with x, they don't swap places? Thank you anyway.
     
  11. Dec 13, 2017 #10

    PeroK

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    The converse of: "All logicians are nerds" is "All nerds are logicians.".
     
  12. Dec 13, 2017 #11

    mfb

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    Is that supposed to be "smaller"? That is (a) wrong and (b) not what the rest of the thread uses.
     
  13. Dec 13, 2017 #12
    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.
     
  14. Dec 13, 2017 #13

    PeroK

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    From your posts on this thread it is difficult to find anything that gives you away as a non-native speaker.
     
  15. Dec 13, 2017 #14
    Not sure if a compliment or a reference that I do not really understand the subject.
     
  16. Dec 13, 2017 #15

    PeroK

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    It was meant as a compliment.
     
  17. Dec 13, 2017 #16
    Thank you. :)
     
  18. Dec 13, 2017 #17

    mfb

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    Well, you used n>x in your post, that lead to my question.
     
  19. Dec 14, 2017 #18
    You are right, I said smaller, but meant greater. Sorry.
     
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