# Set builder notation for this set

I need set builder notation for a set. Set under consideration is:

$$\{m,n,o,p\}$$ What I suggest is:

$$\{x\colon x\in\{m,n,o,p\}\}$$ Any suggestions is it correct?

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Yeah, it is great. Why not $\{x\ |\ x\in\{x\ |\ x\in \{m,n,o,p\}\}\}$. – String Nov 29 '13 at 19:34
Someone who sees it will wonder: why is he not just writing $\left\{ m,n,o,p\right\}$? – drhab Nov 29 '13 at 19:38
If you really have to then you also choose for $\left\{ x\mid x=m\vee x=n\vee x=o\vee x=p\right\}$ – drhab Nov 29 '13 at 19:40
A notation like $\left\{ x\mid x\in A\right\}$ for $A$ doesn't really look good and it isn't surprising that mathematicians make fun of it. The comment of String is an example of that. – drhab Nov 29 '13 at 19:46
@Labeeb: That is ok! The verbal description could be "letters alhabetically from m to p". – String Nov 29 '13 at 21:01

Hint: How would you verbally describe those letters? What do they have in common? Given a verbal description, you can then describe the set with set-builder notation as $$\{x\mid x\text{ [fits verbal description]}\}.$$
For example, I could describe the set $\{\text{red, blue, yellow}\}$ by $$\{x\mid x\text{ is a primary color}\}.$$
If we can't think of a way to describe all of them simultaneously, then no, we can't. However, drhab showed us in the comments above that we can pretty much always do this for finite sets. What do $m,n,o,p$ have in common, though? – Cameron Buie Nov 29 '13 at 19:53