What is the underlying math in this relation?

Question

Suppose we have the constraint $$.7x_1+.4x_2+.5x_3<1,$$ $$x_1,x_2,x_3\in\{0,1\}$$

Then we can convert it to a Boolean expression with binary variables of the form $$(\neg x_{1}\wedge\neg x_{2})\vee\left(\neg x_{1}\wedge\neg x_{3}\right),$$

such that the constraint is satisfied iff the Boolean expression is true.

Could someone explain what math basis this transformation has? I find it very fascinating because we are going from real number coefficients to a logical expressions.

Thanks a lot.

PS. Related to to my previous question Convert a Boolean expression to a linear expression?

Answer 1

Suppose we have n variables. There are only $2^n$ possible true-false combinations, so you can just list out all possible valid combinations, then take their conjunction.