I am wondering about the (very basic) translation of this symbolic set builder notation into natural language:
$$\{n \in \mathbb{Z} : (\exists k \in \mathbb{Z} )[n = 2k]\}$$
It's very intuitively obvious to me that this is the set of all even integers. My English translation would be: "The set of all integers, n, such that there exists an integer, k, such that n is two times k".
I have a number of sub questions from this: (1) Is the translation correct? Specifically is it fine to have the second 'such that'? (2) If the second 'such that; is correct, then why isn't the symbolic notation written as $$\{n \in \mathbb{Z} : (\exists k \in \mathbb{Z}:n = 2k)\}$$
Is there something special about using regular parentheses for $(\exists k \in \mathbb{Z})$, but square parentheses for $[n = 2k]$ that indicates there is a statement 'such that' in between?
Thanks very much.
Your translation is correct. Good job!
The reason there isn't traditionally a second colon in that is because that inner part isn't a set. You could make that part a set if you wrote something like $$\{n\in\mathbb Z:\{k\in\mathbb Z:n=2k\}\neq\emptyset\}$$ but I think the quantifier is more clear about what it's saying.
Different authors have different ways of separating quantifiers like $\exists$ from the logical statement. In your sentence, you used parentheses, which is common. I've also seen people using colons for it, so a reasonable person would have no trouble interpreting your second statement correctly as well. Some of your growth as a mathematician is finding a writing style that captures your personal style of balancing clarity versus brevity, so you're asking the right questions!