Question

Problem 4
Determine |A ∪ B ∪ C| when |A|= 50, |B|= 500, and
|C| 5000, if (a)A ⊆ B ⊆ C; (b)A ∩ B A ∩ C B ∩ C
∅; and (c) |A ∩ B| |A ∩ C| |B ∩ C| 3 and |A ∩ B ∩ C| 1.

Answer 1

Question 1. Determine $|A \cup B \cup C|$ when $|A| = 50$, $|B| = 500$ and $|C| = 5000$, if

(a) $A \subseteq B \subseteq C$;

(b) $A \cap B = A \cap C = B \cap C = \emptyset$;

(c) $|A \cap B| = |A \cap C| = |B \cap C| = 3$ and $|A \cap B \cap C| = 1$.

Solution. (a) If $A \subseteq B \subseteq C$, then $A \cup B \cup C = C$, so $|A \cup B \cup C| = |C| = 5000$.

(b) If $A \cap B = A \cap C = B \cap C = \emptyset$, then $|A \cup B \cup C| = |A| + |B| + |C| = 50 + 500 + 5000 = 5550$.

(c) Now if $|A \cap B| = |A \cap C| = |B \cap C| = 3$ and $|A \cap B \cap C| = 1$, then by inclusion-exclusion formula

$$\begin{array}{l} |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \\ = 50 + 500 + 5000 - 3 - 3 - 3 + 1 \\ = 5542. \end{array}$$

Answer:

(a) 5000;

(b) 5550;

(c) 5542.

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