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

Answer:

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) \}

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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) .

Answer:

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) \}

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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) .

Answer:

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)

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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)

Answer:

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)

Answer:

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) \} 

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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)

Answer:

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 .

Answer:

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.