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$\{x : x \le 75, \in \mathbb{Z}_+ \cup \{0\}\}$

Is this notation correct? I'm going for positive integers less than or equal to $75$ including $0$.

Dominique
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ℤ_INT
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    No, its ${ x\in\mathbb{Z}_{\geq 0} : x\leq 75}$, or ${x\in\mathbb{N}_0: x\leq 75}$. – stange Nov 30 '23 at 12:31
  • Thanks! What notation do I use to convey that the domain of x is that set? – ℤ_INT Nov 30 '23 at 12:34
  • What do you mean by domain of $x$? Domain refers to a function...? – IraeVid Nov 30 '23 at 12:37
  • sorry the domain of f(x) i meant – ℤ_INT Nov 30 '23 at 12:38
  • @stange: really? I always thaught that $\mathbb{N}_0$ means $\mathbb{N}$ without zero, not with zero. – Dominique Nov 30 '23 at 12:38
  • Then you can write "Let $D$ be the domain of $f(x)$". It is the fastest and most convenient way. Otherwise, you can write "${x : f(x)\text{ is defined }}$" which is both ugly and slow. – IraeVid Nov 30 '23 at 12:40
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    @Dominique If you denote with $\mathbb{N}$ the set of natural numbers without $0$, then $\mathbb{N}_0 = \mathbb{N}\cup{0}$, and if $\mathbb{N}$ denotes the set of natural numbers including zero, then $\mathbb{N}^\ast = \mathbb{N}\setminus {0}$. Thats at least the notation I know of. – stange Nov 30 '23 at 12:51
  • @stange: here (in Belgium) it's just the contrary: $\mathbb{N}$ always contains zero, but $\mathbb{N}_0$ does not. – Dominique Nov 30 '23 at 13:02
  • @Dominique, that is quite surprising to me! I haven't ever done maths in Belgium though. Here, here and here all agree with the convention I'm familiar with, that $0 \in \Bbb N_0$. For me notations like $\Bbb N^*$ and $\Bbb Z_{>0}$ would explicitly exclude $0$. I'm not able to find any source at all stating the convention that $0 \notin \Bbb N_0$. Could you point me to one? – Izaak van Dongen Nov 30 '23 at 20:33

1 Answers1

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There are 2 parts to this question. 1) Is your notation correct regarding "mathematical grammar" and 2) Is your notation recommended?

For the first question, "$\{x:x\leq75, \in\mathbb{Z}_+\cup\{0\}\}$" is wrong. You wrote "$\in\mathbb{Z}_+\cup\{0\}$. This is where the error lies. What is $\in\mathbb{Z}_+\cup\{0\}$? You have to write a variable in front. Hence, writing $\{x:x\leq75, x\in\mathbb{Z}_+\cup\{0\}\}$ would be correct regarding "mathematical grammar". You can write like this and people would understand.

For the second question, it is not recommend to write like this. Usually, we write the set that the number belongs to in front. Note that set refers to sets like $\mathbb{N}, \mathbb{R}, \mathbb{Q}$ etc, those that are already defined. Hence, the correct notation will be $\{x\in\mathbb{Z}_{\geq0}:x\leq 75\}$. Or, you can just write $\mathbb{Z}_{\geq0}\cap[0, 75]$.

IraeVid
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