During a discrete mathematics test I got this question. I did not see it in my course and I am baffled, because I don't even know from where to start. All I found on google for such subject were really advanced scientific papers...
So here is the question, I would truly appreciate any answers or hints.
Solve the following system: \begin{cases}abx\leq y, \\ ab(x \vee y)=ax. \end{cases}
Since there are only $4$ possible pairs $(a, b)$ the easiest way to solve this system is to check them all.
If $a = 0$ (the value of $b$ doesn't matter) then we get $y \geq 0$ and $0 = 0$, so any pair $(x, y)$ is the solution.
If $a = 1, b = 0$ we get $y \geq 0$ and $x = 0$, so any pair $(0, y)$ is the solution.
If $a = 1, b = 1$ we get $x \leq y$ and $x \vee y = x$. The latter equation is equivalent to $x \geq y$, which with the former leads to $x = y$, so any pair $(x, x)$ is the solution.
Another approach is to use some Boolean identities and try to obtain the solution in general form. Namely, let's recall that $x \vee y = xy \oplus x \oplus y$. So we can transform our system to the system of polynomial equations over $\mathbb{F}_2$. The original system is equivalent to: $$abxy = abx, \quad abxy \oplus abx \oplus aby = ax.$$ We have $$ax = abxy \oplus abx \oplus aby = abx \oplus abx \oplus aby = aby \Leftrightarrow$$ $$\Leftrightarrow ax \oplus aby = 0 \Leftrightarrow a(x \oplus by) = 0.$$ So either $a = 0$ (and in that case any pair $(x, y)$ is the solution) or $x = by$ (and in that case if $b = 0$ then any pair $(0, y)$ is the solution and if $b = 1$ then any pair $(x, x)$ is the solution) corresponding to the cases above.