Answer to Question #265911 in Discrete Mathematics for nyokie

Question

Given three sets A, B, and C. Suppose the the union of the three sets has cardnality 280. Suppose also that |A| = 100, |B| = 200, and |C| = 150. And suppose we also know |A∩B| = 50, |A∩C| = 80, and |B∩C| = 90. Find the cardinality of the intersection of the three sets.

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

$|A\cap B\cap C|=|A\cup B\cup C|-|A|-|B|-|C|+|A\cap B|+|A\cap C|+|B\cap C|\\ =280-100-200-150+50+80+90\\=50.$