Answer to Question #265874 in Discrete Mathematics for jia

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

Solution:

Given, |A U B U C| = 182, |A| = 92, |B| = 41, |C| = 118. Also, |A ∩ B| = 15, |A ∩ C| = 42, and |A ∩ B ∩ C| = 10

We know that,

|A U B U C| = |A| + |B| + |C| - |A∩B| -|A∩C| - |B∩C| + |A∩B∩C|

Putting given values.

182 = 92 + 41 + 118 -15 -42 -|B∩C| +10

$\Rightarrow$182 = 204 -|B∩C|

$\Rightarrow$|B∩C| = 204 - 182

$\Rightarrow$|B∩C| = 22