# Notation for sum of intersection cardinalities

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?

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: $$\sum_{1 \leq i < j \leq n} |X_i \cap X_j|$$