### Theory:

Let $$A = \{20$$, $$25$$, $$26$$, $$30$$, $$31$$, $$33$$, $$35\}$$, $$B = \{25$$, $$30$$, $$32$$, $$34$$, $$35$$, $$36\}$$ and $$C = \{15$$, $$26$$, $$30$$, $$32$$, $$33$$, $$35\}$$.

(i) Calculate the total number of common elements in all the three sets.

(ii) Calculate the total number of common elements in $$A$$ and $$B$$.

(iii) Calculate the total number of common elements in $$B$$ and $$C$$.

(iv) Calculate the total number of common elements in $$A$$ and $$C$$.

(v) Calculate the total number of elements in all the three sets.

Solution:

$$A = \{20$$, $$25$$, $$26$$, $$30$$, $$31$$, $$33$$, $$35\}$$

Number of elements in set $$A = n(A) = 7$$

$$B = \{25$$, $$30$$, $$32$$, $$34$$, $$35$$, $$36\}$$

Number of elements in set $$B = n(B) = 6$$

$$C = \{15$$, $$26$$, $$30$$, $$32$$, $$33$$, $$35\}$$

Number of elements in set $$C = n(C) = 6$$

(i) Total number of common elements in all the three sets:

$$A \cap B \cap C$$ $$=$$ $$\{30$$, $$35\}$$

$$n(A \cap B \cap C) = 2$$

(ii) Total number of common elements in $$A$$ and $$B$$:

$$A \cap B$$ $$=$$ $$\{25$$, $$30$$, $$35\}$$

$$n(A \cap B) = 3$$

(iii) Total number of common elements in $$B$$ and $$C$$:

$$B \cap C$$ $$=$$ $$\{30$$, $$32$$, $$35\}$$

$$n(B \cap C) = 3$$

(iv) Total number of common elements in $$A$$ and $$C$$:

$$A \cap C$$ $$=$$ $$\{26$$, $$30$$, $$33$$, $$35\}$$

$$n(A \cap C) = 4$$

(v) Total number of elements in all the three sets:

$$n(A \cup B \cup C)$$ $$= n(A) + n(B) + n(C)$$ $$- n(A \cap B)$$ $$- n(B \cap C)$$ $$- n(A \cap C)$$ $$+ n(A \cap B \cap C)$$

$$n(A \cup B \cup C)$$ $$=$$ $$7 + 6 + 6 - 3 - 3 - 4 + 2$$

$$n(A \cup B \cup C)$$ $$= 11$$

Important!
If $$A$$ and $$B$$ are two finite sets, then:

1. $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

2. $$n(A - B) = n(A) - n(A \cap B)$$

3. $$n(B - A) = n(B) - n(A \cap B)$$

4. $$n(A^{\prime}) = n(U) - n(A)$$

5. $$n(U) = n(A) + n(A^{\prime})$$