Let $n \in N$, Is there a difference between:
1) let us assume as true $\exists k \in Z / n= 9 k$ and
2) let us assume as true $ n = 9k / k \in Z$?
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Yes: the second one can be read as : $\forall k (x=9k)$. – Mauro ALLEGRANZA Nov 13 '18 at 15:11
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What difference would it make in case of induction? – Papa Nov 13 '18 at 15:28
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1$∃k (n=9k)$ and $∀k(x=9k)$ are two different statements, with different meanings, irrespective of the contexts where we used them. – Mauro ALLEGRANZA Nov 13 '18 at 15:45
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Is there a way to prove the difference? – Papa Nov 13 '18 at 15:50
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1To prove the difference between "$(n=9k)$ for some $k$" and "$(n=9k)$ for every $k$" ? Try with $n=18$. – Mauro ALLEGRANZA Nov 13 '18 at 15:53
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Thank you so much! – Papa Nov 14 '18 at 10:07
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But this according to you is a proposition : a(n) = (n-1) / n in N, right? – Papa Nov 19 '18 at 05:30
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You appear to be using a nonstandard notation with this "/" symbol. For instance, it is not listed as a logical symbol in Wikipedia, which only lists it as a division or quotient operator, and every other source I could find with a simple Google search revealed the same usage. Anecdotally, this is consistent with my experience as a student of math: the statements that you've written in the question (and in the Nov. 19 comment) parse as invalid.
However, I am going to exercise some judgement here. It seems reasonable to assume that this symbol roughly means "such that":
(1) Let us assume as true $\exists k\in\Bbb Z$ such that $n=9k$.
(2) Let us assume as true $n=9k$ such that $k\in\Bbb Z$.
The second statement still reads a little awkward to me; if I came across it in a research article I would assume it meant
(2*) Let us assume as true $n=9k$ for $k\in\Bbb Z$.
In this case, (1) and (2*) read to me as the same statement. However, both of these are different from another reasonable interpretation of (2):
(2^) Let us assume as true $n=9k$ for all $k\in\Bbb Z$.
In this case, as Mauro mentioned in the comments, (2^) is different from (1), because, for instance: if $n=18$ then $n=9k$ only for $k=2$, but there are other numbers $k\in\Bbb Z$ besides $2$, for which $n=9k$ is not true.
aleph_two
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1If this answers your question, please upvote or give it "best answer" so that it will be removed from the Unanswered Questions list. If not, please edit your question to use more standard notation. – aleph_two Dec 19 '18 at 04:55
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@J.Moh: Hmm, okay. In that case the first sentence strikes me as pretty strange: "There exists $k\in\Bbb Z$ for $n=9k$."? I'm not totally sure why this reads badly to me but my guess is: "X for Y" should be interchangeable with "For Y, [we have] X". But if try that with (1), then you reference $k$ before defining it! – aleph_two Dec 20 '18 at 04:18