Are all the terms in the Inclusion Exclusion Principle Independent of One Another? The terms in the inclusion exclusion principle are simply the (sizes of) the sets themselves and all the possible combinations of $i$ intersections.
And since the intersections of $k$ sets  themselves can be expressed as intersections with itself and the  all the possible combinations of $i$ ($i>k$)  intersections:
e.g 
\begin{multline}X_1∩X_2∩X_3 = (X_1∩X_2∩X_3)' + (X_1∩X_2∩X_3)∩X_4 + \cdots\\ - (X_1∩X_2∩X_3)∩X_4∩X_5 +\cdots \end{multline}
(where $X'$ refers to elements exclusive to $X$)
and $(X_1∩X_2∩X_3)'$ term is exclusive only to $X_1∩X_2∩X_3,$
Then can I say that all the terms in the formula are  independent of each other (i.e can be any whole number), provided that the size of any of the intersections of set $X$  is less than or equal to the size of set $X$?
i.e $|X ∩ S| \leq |X|$,
where $X$ and $S$ can be any set or intersection of sets
 A: Let $I$ be some finite set. For each subset $J\subseteq I$, let $n_J$ be some non-negative integer. Under what conditions can we find a family of sets $(A_i)_{i\in I}$ such that
$$\left|\bigcap_{i\in J} A_i\right|=n_J$$
for all $J$?
The answer is definitely not that it's possible for any family of integers $(n_J)$. First of all, we of course need $n_J\leq n_K$ for $K\subseteq J$, because a family of sets must always be smaller than the intersection of a subfamily of those sets. But that still isn't enough. For example, suppose that for a family of three sets $A, B, C$ we wanted
$$\begin{align}
|A\cap B\cap C|&=1\\
|A\cap B|&=3\\
|A\cap C|&=3\\
|A|&=4
\end{align}$$
The problem is that this implies that $|A\cap B\cap\overline C|$ and $|A\cap C\cap\overline B|$ each contain two elements, which along with the one element $|A\cap B\cap C|$ means $A$ already has $5$ elements. You can try picturing this logic with a Venn diagram.
In fact the values $n_J$ are bound together by a matrix of linear inequations such as
$$|A\cap B|+|A\cap B|-|A\cap B\cap C|\leq |A|$$
One way of teasing out these inequalities is to express everthing in terms of the variables $|\bigcap_i A_i^{x_i}|$, where each $x_i$ is either the set complement operator (relative to $\bigcup_i A_i$) or nothing. These sets are simply the "minimal regions" of a Venn diagram, the parts that you could imagine filling in a single click with the paint bucket tool in a drawing program. These cardinalities are indeed all independent of one another, and the terms of the Inclusion-Exclusion equation can be expressed as linear combinations of them. Inverting that matrix should give you a set of equations with linear combinations of the I-E terms on the right, and the sizes of the minimal regions on the left. A necessary and sufficient condition for a candidate set of a I-E terms to be feasible is then that the values on the left hand side of that system all be non-negative.
