Using set notation to write domains and ranges I am a new mathematics teacher. $\mathbb R$ stands for set of real numbers. I learned that using set notation, a domain can be written $\{X \in \mathbb R\;|\; 4\leq x\leq7\}.$
However, some students write $\{x\;|\;4\leq x\leq7\}$ or $\{x\;|\;x \in \mathbb R, 4\leq x\leq7\}$. Are they correct?
For the domain of $|x|,$ some students write $\{x\;|\;x \in \mathbb R\}$ instead of $\mathbb R.$ Are they correct?
What are other ways people use set notation to write domains and ranges?
 A: I'm not going to claim to be a great authority on this question, but I think judging whether or not these answers are 'correct' depends on what you, as a teacher, are assessing. Generally speaking, the style of specifying the domain is done via set builder notation; where you create a set from some parent set, by first specifying elements of the parent set, and then what logical condition they must satisfy.
For example, if we had some real-valued function $f$ which has a domain constrained to some interval $[a,b]$, then to write out the domain in this set-builder notation we would have $\mathrm{Dom}(f) = \{x\in \mathbb{R} \mid a\leq x \leq b\}$, which would largely be accepted by most people to be clear and unambiguous.
So technically speaking, the answers they have provided does not conform to this traditional style. However, from reading what they have done, it is still very clear what they mean.
Sets like $\{x\mid 4\leq x \leq 7\} $is clear if it is understood that the function we are discussing will always be real valued, for example. Furthermore, $\{x\mid x\in\mathbb{R},  4\leq x \leq 7\}$ is as clear as the usual set builder notation, but not exactly the same syntax.
So as a teacher, it's your choice to judge whether being precise/conforming to notation is more important than being clear and vice versa.
A: When we write $\{x|....\}$ we mean the set of all those & only those $x$ that are named, listed, specified, or defined by whatever "$....$" says. So $R=\{x|x\in R\}$ regardless of what the set $R$ is. (But I suggest asking the students whether they know this!).And $\{x\in R|....\}$ is accepted as meaning the same $\{x|x\in R\land ....\}.$ And using a comma instead of $\land$ is acceptable here as it is not ambiguous or unclear. But $\{x|4\le x\le 7\}$ is incomplete. E.g. $\{x|x\in\Bbb R \land 4\le x\le 7\}$ and $\{x|x\in\Bbb Q\land4\le x\le 7\}$ are different sets.
