# Is my notation correct?

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

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

• No, its $\{ x\in\mathbb{Z}_{\geq 0} : x\leq 75\}$, or $\{x\in\mathbb{N}_0: x\leq 75\}$. Nov 30, 2023 at 12:31
• Thanks! What notation do I use to convey that the domain of x is that set? Nov 30, 2023 at 12:34
• What do you mean by domain of $x$? Domain refers to a function...? Nov 30, 2023 at 12:37
• sorry the domain of f(x) i meant Nov 30, 2023 at 12:38
• @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. Nov 30, 2023 at 12:51

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 $$\mathbb{Z}_{\geq0}\cap[0, 75]$$.