How to write that the sum of two numbers is an element of set {1, 2, .., 8} in set theory?

Question

Is it $\quad a+b\in\{1, 2, ..., 8\}\quad$ or $\quad (a+b)\in\{1, 2, ..., 8\}$?

And is it $\quad S'=S\cup\{a+b\}\quad$ or $\quad S'=S\cup\{(a+b)\}$?

What is the most correct?

Answer 1

It is completely fine to write $a+b \in \lbrace 1,\ldots, 8 \rbrace$. Parentheses are (mostly) used when there is a risk of confusion about the order in which to apply the operations. Since $a + (b \in \lbrace 1,\ldots, 8 \rbrace)$ does not have any meaning whatsoever, it is not necessary to use parentheses. In the same vein, $S' = S \cup \lbrace a + b \rbrace$ is completely fine (and preferable).

Answer 2

Both notations are correct.

I would prefer $$ \quad a+b\in\{1, 2, ..., 8\}\quad$$ and

$$\quad S'=S\cup\{a+b\}\quad$$