Consider sets $A,B,C$ in some space $U$. The sets have common elements shown by the intersections in the Venn's diagram below. Each intersection is unique so the sets $A,B,C,A\cap B, B\cap C,A\cap C$ and $A\cap B\cap C$, abbreviated with A, B, C, AB, BC, AC and ABC, are unique.
I want to understand how to properly use notation over multiple intersections, can I use the following shorthands with the indexing? My proposal is for any amount $n$ of circles
n circles
- Name all parts not in intersection with other components (even the intersections marked in the picture, and define $U$
\begin{equation} A_{i} = A_{i} \, \backslash \cup_{j \neq i} A_{j} \end{equation}
- Intersections (Bjorn)
\begin{equation} A_{i} = \cap_{i=1}^{n} A_{i} \end{equation}
- All other except $\cup_{i} A_{i}$
\begin{equation} U = \cup_{i=1}^{n} A_{i}^{c} \end{equation}
Matrix presentation $A_{s}$ for all of those by the upper triangle
\begin{equation} A_{s} = \begin{bmatrix} A_{11} & A_{12} & A_{13} & \dots & A_{1n} & A_{1(n+1)} \\ 0 & A_{22} & A_{23} & \dots & A_{2n} & A_{2(n+1)} \\ \dots \\ 0 & 0 & 0 & \dots & A_{dn} & A_{d(n+1)}. \\ 0 & 0 & 0 & \dots & 0 & A_{(d+1)(n+1)} \end{bmatrix} \end{equation}
where multi-index values are for intersections by (2). The uni-index values ($A_{ii}$) are about (1) in the diagonal. The $U$ is $A_{dn}$. The matrix proposal for the Venn's diagram where the last column values all except one become zero because the $U$ does not have intersections with $A_{i}$ by (3).
\begin{equation} A_{s} = \begin{bmatrix} A_{11} & A_{12} & A_{13} & \dots & A_{1n} & 0 \\ 0 & A_{22} & A_{23} & \dots & A_{2n} & 0 \\ \dots \\ 0 & 0 & 0 & \dots & 0 & A_{(n+1)(n+1)}. \end{bmatrix} \end{equation}
Case n=3
\begin{equation} A_{s} = \begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 \\ 0 & A_{22} & A_{23} & 0 \\ 0 & 0 & A_{33} & 0 \\ 0 & 0 & 0 & A_{44}. \end{bmatrix} \end{equation}
Definitions
You want to learn different complement operations here. Notice that relative complement and absolute complement are two different things. The absolute complement noted with C, like $A^c$ (all other elements except elements of A i.e. $$A^c=\{x\in U\mid x\not\in A\},$$ while relative complement for sets $A$ and $B$
$$B\backslash A= B\cap A^c$$
where all elements of $B$ not in $A$. The minus sign you use should be a backslash.
The triangle stands for symmetric difference $A\triangle B$, all elements of $A$ and $B$ such that they have no common elements so removing the intersection area.
Answer to the question
Consider the A,B and C picture in the question. The union of the subareas having no intersections are a relative complement of three sets. It could be marked with $$U-A_1\cap A_2-A_1\cap A_3-A_2\cap A_3$$ when you mark the disks with $A_1$, $A_2$ and $A_3$. And the general case is given by Inclusion Exclusion Principle.