Does there exist an equation system with only linear equalities and inequalities and exhibits a NAND-like behavior? By 'NAND-like' I mean, among all the variables in the system there are 3 vars $x_1$, $x_2$, and $x_3$, such that all solutions with $x_3 = 0$ imply $x_1 = x_2 = 1$, and all solutions with $x_3 = 1$ can be classified into 3 categories, with each category receiving at least 1 solution:
$x_1 = 1$, $x_2 = 0$
$x_1 = 0$, $x_2 = 1$
$x_1 = x_2 = 0$
Note: the converse does not need to be hold.