Answer to Question #170980 in Discrete Mathematics for altair

Question

Suppose that 40 Malaysian Chinese are surveyed and they speak at least one of the

three dialects, Cantonese, Hokkien or Hakka. It is found that 30 of them speak

Cantonese, 20 speak Hokkien and 15 speak Hakka. It is also found that 3 of them speak

all three dialects. How many of the surveyed Malaysian Chinese speak exactly two

languages?

Answer 1

Let

A - set of Malaysian Chinese who speak Cantonese;

B - set of Malaysian Chinese who speak Hokkien;

C - set of Malaysian Chinese who speak Hakka;

Then

$|A| = 30,\,\,|B| = 20,\,\,|C| = 15,\,|A \cup B \cup C| = 40,\,\,|A \cap B \cap C| = 3$

Then, by Inclusion–exclusion principle:

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

Then the number of people speaking exactly two languages is

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

$= 30 + 20 + 15 + 3 - 40 = 28$

Answer: 28