All Questions
8
questions
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30
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Elementary question regarding various definitions of the integers and naturals
So the question is as follows
We need to prove that $\mathbb{\mathbb{Z}}$ is the same set as the following three sets
(i) $\{x\in\mathbb{R}\hspace{0.1cm}|\hspace{0.1cm}x\in\mathbb{\mathbb{N}}\hspace{...
- 85
4
votes
1
answer
115
views
Can One Discuss Induction without Sets?
The standard presentation of mathematical induction involves subsets having a certain property. Here is a typical formulation from Gallian's Contemporary Abstract Algebra, Ninth edition:
It seems to ...
- 515
0
votes
2
answers
105
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Is there a specific symbol for $\mathbb{N}\cup\lbrace 0\rbrace$? [duplicate]
It is well known that natural numbers start in 1.
However, sometimes people work with a "widened set" of natural numeres plus zero, $\mathbb{N}\cup\lbrace 0\rbrace$. That is, all non-...
- 164
1
vote
2
answers
71
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How is ($\mathbb{Z}\setminus\mathbb{Q}$) a subset of $\mathbb{N}$?
I do not understand why the set ($\mathbb{Z}\setminus\mathbb{Q}$) is a subset of $\mathbb{N}$. $\mathbb{Q}$ extends the $\mathbb{Z}$ by adding fractions. So there cannot be an element in $\mathbb{Z}$ ...
- 21
3
votes
1
answer
103
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Is there a bijective function $f: \Bbb Z \to \Bbb N$ that involves only elementary arithmetic and no piecewise functions?
As the title suggests, I'm looking for a function $f : \Bbb Z \to \Bbb N$ that satisfies the following:
$$
\forall y \in \Bbb N, \exists! x \in \Bbb Z : y = f(x)
\\
\therefore\quad
\Bbb N = \left\{ f(...
- 227
3
votes
4
answers
2k
views
Natural numbers as a subset of integer numbers: $\mathbb{N}\subset\mathbb{Z}$.
Within set theory, having the natural numbers $\mathbb{N}$ built as the minimal inductive set with the corresponding additive and multiplicative operations defined, integers $\mathbb{Z}$ can be set as ...
- 812
2
votes
3
answers
68
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Linear ordering $\leq$ on $\mathbb{Z}$ in ZFC
Given a set of natural numbers $\mathbb{N}$ in ZFC, we define $\mathbb{Z}$ by first defining an equivalence relation $\simeq$ on $\mathbb{N}\times\mathbb{N}$: $(n,m) \simeq (n',m') \Longleftrightarrow ...
- 4,528
2
votes
1
answer
815
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Prove that the ordering relation $<_Z$ on the integers is well defined.
I am in the beginning of learning set theory, and the author constructs $\mathbb{R}$, $\mathbb{Q}$ and lastly $\mathbb{Z}$, before $\mathbb{N}$.
Definition of $\sim$
The relation $\sim$ between ...
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