Proper notation question?

Question

Are these two ways to define a set both correct?

$$\{Z_t|Z_t \in A_5 \& Z_t=Z_i \cap Z_k, Z_i \in A_2, Z_k\in A_3\}$$

and the same written in this way:

$$\{Z_t|Z_t \in A_5 \& (\exists Z_i \in A_2)(\exists Z_k \in A_3)(Z_t=Z_i \cap Z_k)\}$$

Answer 1

If you connect two logical statements, you must use the according logical operators.

Usually, you do not write a $\&$, but rather a $\wedge$. Although one may use a separation by commas I prefer also using the logical conjunction $\wedge$. Another optional thing is writing the main Set the elements are contained in before the bar.

But a major issue is using $Z_t$ and $Z_k$ before defining them or using them with a quantor:

As far as I did not misunderstand anything, it should be $$ \{ Z_t\in A_5 | \exists Z_i\in A_2 \exists Z_k \in A_3:Z_t=Z_i \cap Z_k \} $$

Which would be more like your second approach.

The major scheme is $\{ x| A(x)\}$ where $A(x)$ is a predicate (See here), which is, roughly said, a logical expression varying under a given parameter. It can be extended to $\{x\in M|A(x)\}:=\{x|x\in M \wedge A(x)\}$

I recommend to you learning about the basics of mathematical logic and its notation, which should help you formulate such statements correctly. A first googling (I recommend “[search terms] filetype:pdf” for pdf documents) gets me this which seems quite good as an introduction.

And to the community, I do not understand why one would vote that question down. Although the content is incorrect, it was the actual question whether it is correct.