# Is there a general and mechanical method to solve algebra of sets (or alg.of propositions) equations?

In some simple cases it seems possible to solve for X a set equation.

For example, if I am given : X Inter U = U , and knowing the law according to which S Inter U = S for any set S, I can find that X = U.

But more generally,do the properties of the algebra of sets/ propositions allow us to solve equations? and to solve them in a mechanical way?

Is it possible to give an example of equation solving where one applies a rule equivalent to " do the same thing on both sides" in ordinary basic algebra.

Remark: by solving a propositional logic equation, I would mean : finding the unknown truth value of a proposition X involved in a given equivalence.

In the case of algebras of sets or boolean algebras however you often can't get "exact" solutions because the operations $$\cap, \cup$$ are far from being as well-behaved as $$\times$$ in a group for instance : you can very easily have $$a\cap b = a\cap c$$ without having $$b=c$$. Therefore when you do the same thing to both sides of an equation, very often you'll only have "original equation implies modified equation", but not the converse (in a group, if you multiply by $$y$$ on both sides, then you get an equivalent equation, which may be easier to solve)
Let's look at your example : $$X\cap U =U$$, without the hypothesis on the $$S$$ you're considering. Then this implies $$U\subset X$$, and conversely, if $$U\subset X$$ then indeed $$X\cap U = U$$. This actually works in any boolean algebra : $$x\land u = u\iff u\leq x$$ : so we have an equation, and the "domain of solutions" is an upward directed set.
In your case, since $$S\cap U = S$$ for all $$S$$, this means $$S\subset U$$ for all $$S$$, i.e. you're only considering subsets of $$U$$, so clearly if $$U\subset X$$ you'll have $$U=X$$ and that's why in this case you can solve to get an "exact" solution, instead of an interval of solutions.