1. ## Research Help

I need some help , i doing my in formal methods (formal verification) i am facing a problem and need an expert opinion i have set say {s | f + g = k} where f and g are some functions
and k is number i want to split this set into two set (union or intersection) as {s | f = i} union (or intersection) {s | g = j} , first i want to ask can i do it , secondly what will be values of i & j (with respect to k), looking forward for some body who can help me out...

2. Clearly, for any i, j such that i + j = k, we have

$\{x\mid f(x)=i\}\cap\{x\mid g(x)=j\}\subset\{x\mid f(x)+g(x)=k\}$.

However, the converse inclusion does not hold because the fact that f(x) + g(x) = k does not uniquely determine f(x) and g(x). The only thing that comes to mind when i, j and k are natural numbers is

$\bigcup_{i=0}^k\{x\mid f(x)=i\}\cap\{x\mid g(x)=k-i\}=\{x\mid f(x)+g(x)=k\}$.

3. Thanks for answering my post