Valid Set builder notations for simple set.

Question

So I have a simple set:

$\{ 1, 10, 100 \}$

I'm pretty sure one other way way I could represent it is:

$\{ 10^{m} : m \in \{0, 1, 2\} \}$

But I have a couple of other ideas for how to represent it. I just want to know if these other ways are possible or if any of these representations don't make sense. Here they are:

$\{ 10^{m} : \forall m \in \{0, 1, 2\} \}$

$\{ 10^{m} : \exists m \in \{0, 1, 2\} \}$

$\{ n : n = 10^{m} \text{ for some } m \text{ in } \{0, 1, 2\} \}$

If some of these are representations are invalid, I would love to know why.

Answer 1

The third one is valid, but the first two are not.

Let's go back to the original $\{ 10^m \mid m \in \{ 0,1,2 \} \}$

This we can read as 'the set of numbers $10^m$ for any $m$ for which it is true that $m \in \{0,1,2 \}$

But try doing that for $\{ 10^m \mid \forall m \ m \in \{ 0,1,2 \} \}$ ... now you get something like 'th set of numbers $10^m$ for any $m$ for which it is true that for all $m$ it is teue that $m \in \{0,1,2 \}$ ... that makes no sense. Same for the one after. The problem is that you need a predicate about $m$ after the $|$ and you don't have that once you quantify $m$, for then $m$ is no longer a free variable.

But the last one works again, for $n = 10^m$ for some $m \in \{0,1,2 \}$ is a predicate about $n$, as desired.

Answer 2

Bram28 has already touched upon the specifics of your question, but I would like to offer an alternative: $$\bigcup_{m=0}^2\bigl\{ 10^m\bigr\} = \bigl\{ 10^0 \bigr\} \cup \bigl\{ 10^1 \bigr\} \cup \bigl\{ 10^2 \bigr\}$$

This denotes that $m$ must take all three values, not that it has to be one of the three values. I personally like this much more.