Answer to Question #265892 in Discrete Mathematics for nyokie

Question

Given three sets A, B, and C. Suppose we know that the union of the three sets has cardinality 182.

Further, |A| = 92, |B| = 41, |C| = 118. Also, |A ∩ B| = 15, |A ∩ C| = 42, and |A ∩ B ∩ C| = 10. Find

|B ∩ C|.

Answer 1

Let us use the inclusion–exclusion principle:

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|.$

It follows that

$|B\cap C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|+|A\cap B\cap C|-|A\cup B\cup C|\\ =92+41+118-15-42+10-182\\=22$