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When I have three variables $x$, $y$, and $z$ and I know that all three are natural numbers that are bigger or equal to $3$, can I write that like this?:

$$x\space\wedge\space y\space\wedge\space z\in\mathbb{N}_3$$

It has to be $x\in\mathbb{N}_3\space\wedge\space y\in\mathbb{N}_3\space\wedge\space z\in\mathbb{N}_3$. It is for a poster, so I want to write it symbolically instead of in words. How can I write it symbolically?

caird coinheringaahing
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2 Answers2

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Write:

$x,y,z$ are natural numbers greater than or equal to $3$.

Don't give in to the desire to write everything in symbols in the hope that it will sound more mathematical like that.

Also, $\mathbb N_3$ is not standard notation. The only version of $\mathbb N$ with a subscript I know is $\mathbb N_0$, meant to indicate that we include $0$, since that's a subject of debate; but nobody argues about whether $3$ is a natural number or not.

Neither is $x \land y \land z$ particularly meaningful. Possibly you mean $\min\{x,y,z\}$ by that, which would be more transparent notation. Or possibly you meant $x \in \mathbb N_3 \land y \in \mathbb N_3 \land z \in \mathbb N_3$? In which case, logical operations don't distribute like that.

Even if someone is 99% certain to be on the same page as you with both of the above... with all the potential for confusion, why not just use words?

If you insist on writing in symbols, I'd write "$x,y,z \in \mathbb N$ and $x,y,z \ge 3$", or at worst "$x,y,z \in \mathbb N_{\ge 3}$". (To be clear: $\mathbb N_{\ge 3}$ is not standard notation either, but it is guessable, because nothing else would be written like that.) If it's understood that $x,y,z$ are natural numbers, then just "$x,y,z \ge 3$" would do.

Misha Lavrov
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  • Thank you for your answer. It has to be $x\in\mathbb{N}_3\space\wedge\space y\in\mathbb{N}_3\space\wedge\space z\in\mathbb{N}_3$. It is for a poster, so I want to write it symbolically instead of in words. How can I write it symbolically? –  Sep 19 '20 at 16:21
  • I've added some suggestions, but I think that even on the poster, words are not a problem. – Misha Lavrov Sep 19 '20 at 16:24
  • I remember comments on this site stating that some people use $\mathbb{N}_0$ to denote $\mathbb{N}\setminus {0}$. I don't recall where, so I can't give a link. – Daniel Fischer Sep 19 '20 at 17:29
  • That... is a good reason not to use $\mathbb N_0$ to denote anything, since I wouldn't even guess that $\mathbb N_0$ is meant to exclude $0$. – Misha Lavrov Sep 19 '20 at 17:30
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    Neither would I have guessed that. The safe option is to always write $\mathbb{N}\cup {0}$ and $\mathbb{N}\setminus {0}$. Each is redundant for half of your readers, but at least it's unambiguous. – Daniel Fischer Sep 19 '20 at 21:05
  • @DanielFischer: are you perhaps thinking of What is the symbol to refer to the set of whole numbers. – Peter Grill Sep 20 '20 at 00:49
  • @PeterGrill No, for me "whole numbers" can sensibly only mean "ganze Zahlen", the symbol for which is $\mathbb{Z}$. And since those who use the term in a different sense cannot agree whether it should denote the nonnegative integers or the positive integers, adding "whole numbers" into the bag only increases the confusion. – Daniel Fischer Sep 20 '20 at 08:53
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You can write it in so many ways, for example:

$x,y,z\in\mathbb{N}$ such that $x\cdot y\cdot z \geq 3\cdot \min\{x\cdot y,x \cdot z, y\cdot z\}$

But why would you? When writing, you want the reader to understand, so write the whole sentence.

Alon Yariv
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