CLASS-8
PROBLEM & SOLUTION OF VENN DIAGRAM

Solved Example –

Let U = { x : x ∈ W and  4 ≤ x < 14 },  A = { x : x is a positive integers and  x ≤ 6 },  B = { x : x is a multiple of 3 }  and C = { even Integers }.

Represent the given sets by a venn diagram and find the following

1)  A ∩ B ,  2)  B ∩ C ,  3) C ∩ A ,  4)   A’ ,  5)   B’ ,  6)   C’ ,   7)    A U B ,   8)   ( A U B )’ ,  9)   ( B U C )’ ,  10)  ( A U C )’


Answer)  We can represent the given sets in the tabular form as U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

We can arrive at the following from the adjoining Venn Diagram which represents the given sets.     


1)  A ∩ B 

Ans.) As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

A ∩ B  = { 4, 5, 6 } ∩ { 6, 9, 12 }  = { 6 }     [ Intersection Set ] (Ans.)


  2)  B ∩ C 

Ans.) As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

B ∩ C =  { 6, 9, 12 } ∩ { 4, 6, 8, 10, 12 } =  { 6, 12 }    (Ans.)



  3)  C ∩ A 

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

C ∩ A  =  { 4, 6, 8, 10, 12 } ∩ { 4, 5, 6 }  =  { 4, 6 }     (Ans.)

 

  4)   A’ 

Ans.)  As per the given condition we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

A’ = U – A = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } – { 4, 5, 6 }

            = { 7, 8, 9, 10, 11, 12, 13 }          (Ans.)

 

  5)   B’ 

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

B’  =  U – B = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } – { 6, 9, 12 }

              = { 4, 5, 7, 8, 10, 11, 13 }      (Ans.)

 

  6)   C’ 

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 } ,  B = { 6, 9, 12 } ,  C = { 4, 6, 8, 10, 12 }

C’ = U – C = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } - { 4, 6, 8, 10, 12 }     =  { 5, 7, 9, 11, 13 }       (Ans.)



  7)   A U B

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

 A U B = { 4, 5, 6 } U { 6, 9, 12 }  =  { 4, 5, 6, 9, 12 }     (Ans.)


 

   8)   ( A U B )’

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

( A U B )’ =  U - ( A U B ) 

          =  { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } – { 4, 5, 6, 9, 12 }

          =   { 7, 8, 10, 11, 13 }             (Ans.)

 

 9)   ( B U C )’

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 } ,  B = { 6, 9, 12 } ,  C = { 4, 6, 8, 10, 12 }

( B U C )  =  { 6, 9, 12 } U { 4, 6, 8, 10, 12 } 

            =  { 4, 6, 8, 9, 10, 12 }


 ( B U C )’  =   U - ( B U C ) 

   =   { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } – { 4, 6, 8, 9, 10, 12 }

   =    { 5, 7, 11, 13 }             (Ans.)

 

                                                                                       

         

  10)  ( A U C )’

Ans.)  As per the given condition, we have -

U = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

A = { 4, 5, 6 },  B = { 6, 9, 12 },  C = { 4, 6, 8, 10, 12 }

    A U C  =  { 4, 5, 6 } U { 4, 6, 8, 10, 12 }

            =  { 4, 5, 6, 8, 10, 12 }

 ( A U C )’ =  U - ( A U C ) 

      = { 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 } - { 4, 5, 6, 8, 10, 12 }

       =  { 7, 9, 11, 13 }         (Ans.)