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

Answer:2019-04-21_12-19-55

(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

\\

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)

Answer:

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 \} 

\\

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

Answer:

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 \}

\\

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

Answer:

(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 \}  

\\

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)'

Answer:

(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 \}

\\

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'

Answer:

(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'

\\