Quantifier in Set definitions

Question

Can the definition be made more readable:

$\overline{R_1} = \{ (j_1,j_2)\mid j_2 < j_1 + \Gamma^r_a + c^r_w \text{ and } j_1 \in J^r_v \text{ and } j_2 \in J^r_v \text{ and } a = (v,w) \in A^r \text{ and } w \neq D^r \text{ for some } r \in R\}$

Answer 1

What you’re written doesn’t quite make sense. I suspect that what you want is

$$\left\{\left(j^{r_1},j^{r_2}\right):\exists w\in R\setminus\{D\}\,\big((r_1,r_2)\in w\text{ and }j^{r_1}<j^{r_2}\big)\right\}$$

or, without the quantifier but with more words,

$$\left\{\left(j^{r_1},j^{r_2}\right):j^{r_1}<j^{r_2}\text{ and }(r_1,r_2)\in w\text{ for some }w\in R\setminus\{D\}\right\}\;.$$

This can also be written

$$\left\{\left(j^{r_1},j^{r_2}\right):(r_1,r_2)\in\bigcup\left(R\setminus\{D\}\right)\text{ and }j^{r_1}<j^{r_2}\big)\right\}\;,$$

since $(r_1,r_2)\in\bigcup\left(R\setminus\{D\}\right)$ if and only if there is a $w\in R\setminus\{D\}$ such that $(r_1,r_2)\in w$.

Added: For your set $\overline{R_1}$ you can write

$$\overline{R_1}=\left\{(j_1,j_2):\exists r\in R\,\exists a=(v,w)\in A^r\Big(w\ne D^r\mathrel{\&}j_1,j_2\in J_v^r\mathrel{\&}j_2<j_1+\Gamma_a^r+c_w^r\Big)\right\}\;.$$