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If $A,B,C$ are sets, then we can say that the sum of the cardinality intersections is $|A \cap B|+|B \cap C| + |C \cap A|$. However, with more sets, this may become tedious. I understand that there is this notation with the large intersection: $\bigcap$, but this cycles through intersections rather than sums of cardinalities of intersections.

I wonder if something like the following cyclic sum notation is common, if $p = (A,B,C)$ is a permutation of the three sets: $$\sum_{p}A\cap B=|A\cap B| + |B \cap C| + |C \cap A|.$$

If it is not the standard (which it probably is not, because it is a bit ambiguous and I have not seen it before), what is the preferred notation for what I am trying to state? Also, how do we generalize this notation beyond two intersections?

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There's a notation that's similar. I prefer this: Let $X = \{ X_1, X_2, \cdots, X_n \}$ be a family of sets. We can instead sum over all pairs $i \neq j$ in the following manner: $$ \sum_{1 \leq i < j \leq n} |X_i \cap X_j| $$

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