Question 1: Given $A =\{1,2,3\}$ , $B = \{3,4\}$ , $C =\{4,5,6\}$ , find $(A \times B) \cap (B \times C)$.

Given $A =\{1,2,3\}$ , $B = \{3,4\}$ , $C =\{4,5,6\}$

$A \times B = \{ 1,2,3 \} \times \{ 3,4 \} = \{ (1,3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4) \}$

$B \times C = \{3,4\} \times \{4,5,6\} = \{ (3, 4), (3,5), (3, 6), (4, 4), (4,5), (4, 6) \}$

$\therefore (A \times B) \cap (B \times C)= \{ (3, 4) \}$

$\\$

Question 2: If $A= \{ 2,3 \}$ , $B = \{4,5\}$ , $C = \{5, 6 \}$ , find $A \times (B \cup C)$ , $A\times (B \cap C)$ , $(A \times B) \cup (A \times C)$ .

Given $A= \{ 2,3 \}$ , $B = \{4,5\}$ , $C = \{5, 6 \}$

$\therefore B \cup C = \{ 4, 5, 6 \}$

$A \times (B \cup C) = \{ 2,3 \} \times \{ 4, 5, 6 \} = \{ (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6) \}$

Now $B \cap C = \{5 \}$

$\therefore A \times ( B \cap C) = \{ 2, 3 \} \cap \{ 5 \} = \{ (2, 5), (3, 5) \}$

Now $A \times B = \{ 2, 3 \} \times \{4,5\} = \{ (2, 4), (2, 5), (3, 4), (3, 5) \}$

and $A \times C = \{ 2, 3 \} \times \{5, 6\} = \{ (2, 5), (2, 6), (3, 5), (3, 6) \}$

$\therefore (A \times B) \cup (A \times C) = \{ (2, 4), (2, 5), (2, 6), (3, 4), (3,5 ), (3, 6) \}$

$\\$

Question 3: If $A = \{ 1, 2, 3 \}$ , $B = \{ 4 \}$ , $C = \{ 5 \}$ , then verify that: (i) $A \times (B \cup C)=(A \times B) \cup (A \times C)$   (ii) $A \times (B \cap C)=(A \times B) \cap (A \times C)$   (iii) $A \times (B-C) =(A \times B) - (A \times C)$ .

i)    Given $A = \{ 1, 2, 3 \}$ , $B = \{ 4 \}$ , $C = \{ 5 \}$

$\therefore B \cup C = \{ 4 \} \cup \{ 5 \} = \{4, 5 \}$

$\therefore A \times (B \cup C) = \{ 1, 2, 3 \} \times \{4, 5 \} = \{ (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) \}$   … i)

Now $A \times B = \{ 1, 2, 3 \} \times \{ 4 \} = \{ (1, 4), (2, 4), (3, 4) \}$

and $A \times C = \{ 1, 2, 3 \} \times \{ 5 \} = \{(1, 5), (2, 5), (3, 5) \}$

$\therefore (A \times B) \times (A \times C) = \{ (1, 4), (2, 4), (3, 4) \} \cup \{(1, 5), (2, 5), (3, 5) \} \\ \hspace*{0.8cm} = \{ (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) \}$  … ii)

From i) and ii) we get

$A \times (B \cup C)=(A \times B) \cup (A \times C)$. Hence Proved.

ii)  Given $A = \{ 1, 2, 3 \}$ , $B = \{ 4 \}$ , $C = \{ 5 \}$

$\therefore B \cap C = \{ 4 \} \cap \{5 \} = \phi$

$\therefore A \times (B \cap C) = \{ 1, 2, 3 \} \times \phi = \phi$   … … … … … i)

Now $A \times B = \{ 1, 2, 3 \} \times \{ 4 \} = \{ (1, 4), (2, 4), (3, 4) \}$

$A \times C = \{ 1, 2, 3 \} \times \{ 5 \} = \{ (1, 5), (2, 5), (3, 5) \}$

$\therefore (A \times B ) \cap (A \times C) = \{ (1, 4), (2, 4), (3, 4) \} \cap \{ (1, 5), (2, 5), (3, 5) \} = \phi$ … ii)

From i) and ii) we get  $A \times (B-C) =(A \times B) - (A \times C)$

iii) Given $A = \{ 1, 2, 3 \}$ , $B = \{ 4 \}$ , $C = \{ 5 \}$

$\therefore B - C = \{ 4 \} - \{ 5 \} = \{ 4 \}$

$A \times ( B - C) = \{ 1, 2, 3 \} \times \{ 4 \} = \{ (1, 4), (2, 4), (3, 4) \}$   … … … … … i)

Now $A \times B = \{ 1, 2, 3 \} \times \{ 4 \} = \{ (1, 4), (2, 4), (3, 4) \}$

and $A \times C = \{ 1, 2, 3 \} \times \{ 5 \} = \{ (1, 5), (2, 5), (3, 5) \}$

$\therefore (A \times B) - (A \times C) = \{ (1, 4), (2, 4), (3, 4) \} - \{ (1, 5), (2, 5), (3, 5) \} \\ \hspace*{4.0cm} = \{ (1, 4), (2, 4), (3, 4) \}$  … ii)

From i) and ii) we get  $A \times (B-C) =(A \times B) - (A \times C)$

$\\$

Question 4: Let $A = \{ 1, 2 \}$ , $B = \{1, 2, 3, 4\}$ , $C = \{ 5, 6 \}$ and $D = \{ 5, 6,7, 8 \}$ . Verify that: (i) $A \times C \subset B \times D$   (ii) $A \times (B \cap C) =(A \times B) \cap (A \times C)$

i) Given $A = \{ 1, 2 \}$ , $B = \{1, 2, 3, 4\}$ , $C = \{ 5, 6 \}$ and $D = \{ 5, 6,7, 8 \}$

$\therefore A \times C = \{ 1, 2 \} \times \{ 5, 6 \} = \{ (1, 5), (1, 6), (2, 5), (2, 6) \}$ … … i)

$B \times D = \{1, 2, 3, 4\} \times \{ 5, 6,7, 8 \} = \{ (1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), \\ \hspace*{2.0cm} (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8) \}$ … ii)

From i) and ii) we see $A \times C \subset C \times D$

ii) Given $A = \{ 1, 2 \}$ , $B = \{1, 2, 3, 4\}$ , $C = \{ 5, 6 \}$ and $D = \{ 5, 6,7, 8 \}$

$\therefore B \cap C = \{1, 2, 3, 4\} \cap \{ 5, 6 \} = \phi$

$\therefore A \times (B \cap C) = \{ 1, 2 \} \times \phi = \phi$ … … …i)

Now, $A \times B = \{ 1, 2 \} \times \{1, 2, 3, 4\} = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) \}$

and $A \times C = \{1, 2 \} \times \{ 5, 6 \} = \{ (1, 5), (1, 6), (2, 5), (2,6) \}$

$\therefore (A \times B) \cap (A \times C) = \phi$   … … … … … ii)

From i) and ii) we get $A \times (B \cap C) =(A \times B) \cap (A \times C)$

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Question 5: If $A = \{1, 2, 3\}$ , $B =\{3, 4\}$ and $C = \{4,5,6\}$ , find
(i) $A \times (B \cap C)$   (ii) $(A \times B) \cap (A\times C)$   (iii) $A \times (B \cup C)$ (iv) $(A \times B) \cup (A \times C)$

i) Given $A = \{1, 2, 3\}$ , $B =\{3, 4\}$ and $C = \{4,5,6\}$

$\therefore B \cap C = \{3, 4\} \cap \{4,5,6\} = \{ 4 \}$

$\therefore A \times (B \cap C) = \{1, 2, 3\} \times \{ 4 \} = \{ (1, 4), (2, 4), (3, 4) \}$

ii) Given $A = \{1, 2, 3\}$ , $B =\{3, 4\}$ and $C = \{4,5,6\}$

$\therefore A \times B = \{1, 2, 3\} \times \{3, 4\} = \{ (1, 3), (1, 4), (2, 3), (2, 4) ,(2, 3), (2, 4) \}$

$A \times C = \{1, 2, 3\} \times \{4,5,6\} = \{ (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6) \}$

$\therefore (A \times B) \cap (A \times C) = \{(1, 4), (2, 4), (3, 4) \}$

iii) Given $A = \{1, 2, 3\}$ , $B =\{3, 4\}$ and $C = \{4,5,6\}$

$\therefore B \cup C = \{3, 4\} \cup \{4,5,6\} = \{3, 4, 5, 6 \}$

$\therefore A \times ( B \cup C) = \{1, 2, 3\} \times \{3, 4, 5, 6 \} \\ \hspace*{0.2cm} = \{ (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6),(3, 3), (3, 4), (3, 5), (3, 6) \}$

iv) Given $A = \{1, 2, 3\}$ , $B =\{3, 4\}$ and $C = \{4,5,6\}$

$\therefore A \times B = \{1, 2, 3\} \times \{3, 4\} = \{ (1, 3), (1, 4), (2, 3), (2, 4), (3, 3) , (3, 4) \}$

$A \times C = \{1, 2, 3\} \times \{4,5,6\} \\ \hspace*{1.0cm} = \{ (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6) \}$

$\therefore (A \times B) \cup (A \times C) = \{ (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), \\ \hspace*{4.5cm} (3, 3), (3, 4), (3, 5), (3, 6) \}$

$\\$

Question 6: Prove that(i) $(A \cup B) \times C=(A \times C) \cup (B \times C)$ (ii) $(A \cap B) \times C =(A \times C) \cap (B\times C)$

i)    Let $(a, b)$ be any arbitrary elements of $(A \cup B) \times C$

$\therefore (a, b) \in ( A \cup B) \times C$

$\Rightarrow a \in (A \cup B)$ and $b \in C$

$\Rightarrow (a \in A \ or \ a \in B)$ and $b \in C$

$\Rightarrow (a \in A \ and \ b \in C)$ or $( a \in B \ and \ b \in C)$

$\Rightarrow (a, b) \in ( A \times C )$ or $( a, b) \in ( B \times C)$

$\Rightarrow (a, b) \in ( A \times C) \cup ( B \times C)$

$\therefore ( A \cup B ) \times C \subseteq ( A \times C) \cup ( B \times C)$  … … i)

Let $(x, y)$ be any arbitrary elements of $(A \times C) \cup (B\times C)$

$\therefore (x, y) \in ( A \times C ) \cup ( B \times C)$

$\Rightarrow (x, y) \in ( A \times C )$ or $(x, y) \in ( B \times C)$

$\Rightarrow (x \in A \ and \ y \in C)$ or $( x \in B \ and \ y \in C)$

$\Rightarrow ( x \in A \ or \ x \in B)$ or $y \in C$

$\Rightarrow ( x \in A \cup B)$ and $y \in C$

$\Rightarrow (x, y) \in (A \cup B ) \times C$

$\therefore ( A \times C) \cup ( B \times C) \subseteq ( A \cup B ) \times C$ … … ii)

From i) and ii) we get $(A \cup B) \times C=(A \times C) \cup (B \times C)$

ii) Let $(a, b)$ be any arbitrary elements of $(A \cap B) \times C$

$\therefore (a, b) \in ( A \cup B) \times C$

$\Rightarrow a \in (A \cup B)$ and $b \in C$

$\Rightarrow (a \in A \ and \ a \in B)$ and $b \in C$

$\Rightarrow (a \in A \ and \ b \in C)$ and $( a \in B \ and \ b \in C)$

$\Rightarrow (a, b) \in ( A \times C )$ and $( a, b) \in ( B \times C)$

$\Rightarrow (a, b) \in ( A \times C) \cup ( B \times C)$

$\therefore ( A \cup B ) \times C \subseteq ( A \times C) \cup ( B \times C)$  … … i)

Let $(x, y)$ be any arbitrary elements of $(A \times C) \cap (B\times C)$

$\therefore (x, y) \in ( A \times C ) \cup ( B \times C)$

$\Rightarrow (x, y) \in ( A \times C )$ and $(x, y) \in ( B \times C)$

$\Rightarrow (x \in A \ and \ y \in C)$ and $( x \in B \ and \ y \in C)$

$\Rightarrow ( x \in A \ and \ x \in B)$ and $y \in C$

$\Rightarrow ( x \in A \cup B)$ and $y \in C$

$\Rightarrow (x, y) \in (A \cup B ) \times C$

$\therefore ( A \times C) \cup ( B \times C) \subseteq ( A \cup B ) \times C$ … … ii)

From i) and ii) we get $(A \cap B) \times C=(A \times C) \cap (B \times C)$

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Question 7: If. $A \times B \subseteq C \times D$ and $A \times B = \phi$ , prove that $A \subseteq C$ and $B \subseteq D$.

Let $(a, b)$ be any arbitrary elements of $(A \times B)$. Then

$(a, b) \in ( A \times B)$

$\Rightarrow a \in A$ and $b \in B$ … … … i)

Now, $(a, b) \in A \times B$

$\Rightarrow (a, b) \in C \times D \hspace*{2.0cm} [\because A \times B \subseteq C \times D ]$

$\Rightarrow a \in C$ and $b \in D$ … … … ii)

$\therefore a \in A \Rightarrow a \in C \Rightarrow A \subseteq C$

and $b \in B \Rightarrow b \in D \Rightarrow B \subseteq D$

Hence Proved.