I have a set of sets $G = \{D_{0,0}\,D_{0,1}\,D_{0,2}\,D_{1,0},…,D_{n,0}\,D_{n,m}\} $
What I know want to express is a constraint that for each set in $ G $, if $ x \in D_{0,0} $ then the statement $ y \in D_{0,1} $ can't be true. So basically, if the set with index 0 has the element $x$ in it, the set with index 1 is not allowed to have an element y. Now the problem is, that I want to have this constraint for all the possible combinations.
$ D_{0,0} $ and $ D_{0,1} $, $ D_{0,1} $ and $ D_{0,2} $, $ D_{1,0} $ and $ D_{1,1} $, and so on. In programming this would basically be a nested for loop. But how would I express something like that in Mathematics?
What I have so far is this: I count all the violations, and if they are 0, it is correct. Is this possible in Mathematics?
$\sum_{i = 1}^{n} \sum_{j = 1}^{m-1} x \in D_{i,j}\wedge y \in D_{i,j+1})=0 $
Best Answer
One thing that's a bit unfortunate is that for two statements $P$ and $Q$, $P \wedge Q$ denotes another statement and not a number like $0$ and $1$, so trying to sum such things to get a number is problematic (compare it with programming languages where boolean True and 1 are different). Now you could augment your summand with a function which you define to take the values 0 and 1 on false and true statements respectively, but that's maybe a little weird.
In your situation, you're in luck though: implementing the constraint of elements belong to certain sets is so common that there's standard notation for it, the underlying notion being that of an indicator function: if $A$ is a subset of some set $X$, then the indicator function $1_A$ takes the value $1$ on elements of $A$ and $0$ on elements in $X \setminus A$.
Suppose now that you have two elements $x , y \in X$ and two subsets $A$ and $B$ of $X$. Then the product $1_A(x)1_B(y)$ is equal to $1$ if both $x \in A$ and $y \in B$, and otherwise it's zero.