To calculate the intersection between three sets, A, B, and C, you can use the inclusion-exclusion principle. We assume you will know all the quantities mentioned.

1. Find the cardinality of the union of all three sets (|A ∪ B ∪ C|).
2. Subtract the cardinality of every single set(|A|, |B|, and |C|).
3. Add the cardinality of the intersection of each pair of set (|A ∩ B|, |A ∩ C|, and |B ∩ C|).

The result will be the cardinality of the intersection |A ∩ B ∩ C|.

The symmetric difference of two sets, A and B, is the set that contains all the elements belonging exclusively to either A or B. The corresponding logical operation is the exclusive or: when both sets exist in a given portion of the diagram, we disregard it. With higher numbers of sets in your diagram, use sum modulo 2 of the overlaps of the sets. If the overlaps are even, exclude the subset. If the overlaps are odd, include the subset.

The inclusion-exclusion principle is a fundament of set theory, a formula that allows you to calculate the union or intersection of any number of sets. To calculate the union, follow these easy steps:

1. Sum the cardinalities of the individual sets.
2. Subtract the cardinality of the intersections of each pair of sets.
3. Sum the cardinality of each intersection of three sets.

This alternate fashion of sum and difference repeats until you end up with possible intersections. This operation results in the cardinality of the union of the sets.