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 union of A and B, with |A| = 10 and |B| = 12 is |A ∪ B| = 18. To find this result, we use the inclusion-exclusion principle:

1. Calculate the sum of the cardinalities |A| and |B|: |A| + |B| = 10 + 12 = 22.
2. Subtract the cardinality of their intersection: |A| + |B| - |A ∩ B| = 22 - 4 = 18.

That's it. We subtracted the intersection because, by summing |A| and |B|, we inadvertently counted twice the elements shared by the two sets.