Is it correct to say that the natural numbers are a proper subset of the integers? $\mathbb{N} \subset \mathbb{Z}$.
Just want to be absolutely sure.
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5God, no way. The integers contain all sorts of GMO numbers and the like. Stick with organic numbers. – copper.hat Mar 29 '18 at 05:03
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6@copper.hat I have flagged your comment for moderator attention. There is no scientific evidence that GMO numbers are inferior to organic numbers. – The Pointer Mar 29 '18 at 05:05
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Yes, but still $\Bbb Z$ has the same cardinality as $\Bbb N$. So there is a bijection between the two sets. – N74 Mar 29 '18 at 06:18
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...but the bijection is not induced by the subset inclusion. – Tyrone Mar 29 '18 at 09:53
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Actually, if you want to be really precise about it, strictly speaking ℕ is NOT a subset of ℤ.
The reason for this is that, when constructing the integers, we define them as equivalence classes and therefore the positive integers are not the same kind of mathematical objects as the natural numbers.
But, since it can be shown that N is order isomorphic to the positive Integers, then for all practical purposes you can treat ℕ as if it was in fact a subset of ℤ.
Erdös
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Undoubtedly N is subset of Z, since atleast we.are in a position to find atleast one element which is there Z but not in N and. Even all the elements of N lies in Z – nimmy Nov 08 '18 at 08:39
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@nimmy, there certainly is a bijection between the elements of N and the positive Integers and they do behave precisely the same way, but, as I said, they are not the same kind of mathematical objects, but you could say the following not-so-formal statement: For any natural number n, there exists some positive integer z such that n and z behave the same way with regards to their ordering and some operation. But, to be clear, this distinction is really irrelevant for all practical purposes. It only comes up when trying to formally construct the integers. – Erdös Nov 09 '18 at 12:52
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@Erdös: you are appealing to an implementation detail: see my comment on https://math.stackexchange.com/questions/4920940/is-mathbb-n-formally-not-a-subset-of-mathbb-z#4920940 – Rob Arthan May 22 '24 at 20:47
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Yes, you are correct.
The natural numbers are subset of integers.
However, the natural numbers do not include any negative integer.
Siong Thye Goh
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Yes. Integers are the essentially the natural numbers and their opposites, plus zero.
Since $\Bbb Z$ contains one or more element not found in $\Bbb N$ (namely $0$ and the negative numbers) and all elements of $\Bbb N$ are found in $\Bbb Z$, then $\Bbb N$ is a proper subset of $\Bbb Z$.
Andrew Li
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Yes: the two sets are not equal, and for any $n \in \mathbf{N}$, we have $n \in \mathbf{Z}$.
Sam Cassidy
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