Giving the range in the natural numbers

Question

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?

Answer 1

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.

Answer 2

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.