I'm trying to say set A is the set of nonnegative integers that not of this two forms $3x^2 + (6y-4)x - y\ $ and $\ 3x^2 + (6y-2)x + (y - 1)$,
for example: $4=3 \cdot1^2+(6 \cdot1-4) \cdot 1-1\ $ is the minimum number of the tow forms, so that $\ 0, \ 1, \ 2,\ 3\ $ must not be of the two forms,then $\ 0, \ 1, \ 2,\ 3\ $ is elements of $A$.
However I got this question: If for each $1 \leq i \leq 8$ I define $X_i$ to be the image of $\mathbb{Z}_{\geq 1} \times \mathbb{Z}_{\geq 1}$ under $f_i$, are you defining $A$ to be the complement of $X_1 \cup X_2$ in $\mathbb{Z}_{\geq 1}$, or the complement of $X_1 \cap X_2$?
(Start of questionable notation)
Let $x,y\in \mathbb{Z}^{+}$,we denote eight polynomials by $f_i(x,y)$ for $i=1, 2, ... ,8$, respectively. $$\begin{cases} f_1(x,y) = 3x^2 + (6y-4)x - y \\ f_2(x,y) = 3x^2 + (6y-2)x + (y - 1) \\ \vdots \end{cases} $$ Let $a$ be nonnegative integer. $$\begin{cases} A = \{a\ not\ of\ f_1\ and\ f_2\} \\ B = \{a\ not\ of\ f_3\ and\ f_4\} \\ \vdots \end{cases} $$ (End of questionable notation)
How to define these set correctly?