Question 1: If $A$ and $B$ are two sets such that $A \subset B$ then find:  (i) $A \cap B$    (ii)  $A \cup B$

(i) Since $A \subset B$, every element of A is in B. Hence $A \cap B$ is nothing bu $A$.

(ii) Since B contains all the elements of A, the $A \cup B$ is nothing but $B$

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Question 2: If $A = \{ 1, 2, 3, 4, 5 \}$, $B = \{ 4, 5, 6,7, 8 \}$, $C = \{ 7, 8, 9, 10, 11 \}$ and $D = \{ 10, 11, 12, 13, 14 \}$. Find:  (i) $A \cup B$  (ii) $A \cup C$  (iii) $B \cup C$  (vi) $B \cup D$   (v) $A \cup B \cup C$  (vi) $A \cup B \cup D$  (vii) $B\cup C \cup D$  (viii) $A \cap (B \cup C)$   (ix) $(A \cap B) \cap (B \cap C)$   (x) $(A \cup D) \cap (B \cup C)$

Given: $A = \{ 1, 2, 3, 4, 5 \}$, $B = \{ 4, 5, 6,7, 8 \}$, $C = \{ 7, 8, 9, 10, 11 \}$ and $D = \{ 10, 11, 12, 13, 14 \}$

(i) $A \cup B = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$

(ii) $A \cup C = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \}$

(iii) $B \cup C = \{ 4, 5, 6, 7, 8, 9, 10, 11 \}$

(vi) $B \cup D = \{ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}$

(v) $A \cup B \cup C = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \}$

(vi) $A \cup B \cup D = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}$

(vii) $B\cup C \cup D = \{ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}$

(viii) $A \cap (B \cup C) = \{ 4,5 \}$

(ix) $(A \cap B) \cap (B \cap C) = \phi$

(x) $(A \cup D) \cap (B \cup C) = \{ 4, 5, 10, 11 \}$

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Question 3: Let $A = \{x: x \in N \}, B = \{ x : x = 2n , n \in N \}$ , $C = \{ x : x = 2n-1 , n \in N \} , D = \{ x : x$ is a prime natural number $\}$ Find:  (i) $A \cap B$  (ii) $A \cap C$  (iii) $A \cap D$  (iv) $B \cap C$  (v) $B \cap D$  (vi) $C \cap D$

$A = \{x: x \in N \} = \{ 1, 2, 3, 4, 5, \cdots \}$

$B = \{ x : x = 2n , n \in N \} = \{ 2, 4, 6, 8, 10, \cdots \}$

$C = \{ x : x = 2n-1 , n \in N \} = \{ 1, 3, 5, 7, 9, \cdots \}$

$D = \{ x : x$ is a prime natural number $\} = \{ 2, 3, 5, 7, 11, 13, 17, 23, \cdots \}$

(i) $A \cap B = \{ 2, 4, 6, 8, 10, \cdots \} = B$

(ii) $A \cap C = \{ 1, 3, 5, 7, 9, \cdots \} = C$

(iii) $A \cap D = \{ 2, 3, 5, 7, 11, 13, \cdots \} = D$

(iv) $B \cap C = \phi$

(v) $B \cap D = \{ 2 \}$

(vi) $C \cap D = \{ 3, 5, 7, 11, 13, 17, 23, \cdots \} = D - \{2 \}$

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Question 4: Let $A = \{ 3,6,12,15,18,21\}, B= \{ 4,8,12,16,20 \}$ , $C = \{ 2,4,6,8,10,12,14,16 \}$ and $D = \{ 5,10,15,20 \}$. Find: (i) $A - B$  (ii) $A - C$    (iii) $A - D$    (iv) $B - A$  (v) $C - A$  (vi) $D - A$  (vii) $B - C$  (viii) $B - D$

(i) $A - B = \{ 3, 6, 15, 18, 21 \}$

(ii) $A - C = \{ 3, 15, 18, 21 \}$

(iii) $A - D = \{ 3, 6, 12, 18, 21 \}$

(iv) $B - A = \{ 4, 8, 16, 20 \}$

(v) $C - A = \{ 2, 4, 8, 10, 14, 16 \}$

(vi) $D - A = \{ 5, 10, 20 \}$

(vii) $B - C = \{ 20 \}$

(viii) $B - D = \{ 4, 8, 12, 16 \}$

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Question 5: If $U = \{ 1, 2, 3, 4,5 ,6 ,7,8, 9 \}$ , $A = \{ 1, 2 ,3 ,4 \}$ , $B = \{ 2, 4, ,6 ,8 \}$ and $C = \{ 3, 4, ,5,6 \}$ Find:  (i) $A'$   (ii) $B'$    (iii) $(A \cap C)'$   (iv) $(A \cup B)'$  (v) $(A')'$   (vi) $(B-C)'$

(i) $A' = \{ 5, 6, 7, 8, 9 \}$

(ii) $B' = \{ 1, 3, 5, 7, 9 \}$

(iii) $(A \cap C)' = \{ 1, 2, 5, 6, 7, 8, 9 \}$

(iv) $(A \cup B)' = \{ 5, 7, 9 \}$

(v) $(A')' = \{ 1, 2 ,3 ,4 \} = A$

(vi) $(B-C)' = \{ 1, 3, 4, 5, 6, 7, 9 \}$

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Question 6: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9 \}, A = \{2, 4, 6, 8 \}$ and $B = \{ 2, 3, 5, 7\}$. Verify that:  (i) $(A \cup B)' = A' \cap B'$   (ii) $(A \cap B)' = A' \cup B'$

(i)  $A \cup B = \{ 2, 3, 4, 5, 6, 7, 8 \}$

$\Rightarrow (A \cup B)' = \{ 1, 9 \}$

$A' = \{ 1, 3, 5, 7, 9 \}$

$B' = \{ 1, 4, 6, 8, 9 \}$

$A' \cap B' = \{ 1, 9 \}$

Hence $(A \cup B)' = A' \cap B'$

(ii)  $A \cap B = \{ 2 \}$

$\Rightarrow (A \cap B)' = \{ 1, 3, 4, 5, 6, 7, 8, 9 \}$

$A' = \{ 1, 3, 5, 7, 9 \}$

$B' = \{ 1, 4, 6, 8, 9 \}$

$A' \cup B' = \{ 1, 3, 4, 5, 6, 7, 8, 9 \}$

Hence $(A \cap B)' = A' \cup B'$

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