Suppose Set A contains 19 elements and Set B contains 69 elements. If the total number elements in either Set A or Set B is 75, how many elements do Sets A and B have in common?

B u A = 48

B = 45

The part of set B not included in A is B - A∩B  = 45 - 7 = 38

Since B u A = 48,  A must have 48-38 = 10 elements (3 exclusive of B, 7 shared with B).

Alternatively, the inclusion-exclusion principle may be used:   A  +  B  - (A ∩ B) = A u B  

  A + 45 - 7 = 48

  A  = 48 - 45 + 7  

   A = 10

NOTE that I'm using A and B  to indicate the number of elements in each set, respectively.

More formally, one would write |A|  and |B| to indicate the cardinality (number of elements) of the set.

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