Complement of a Set

     The complement of a set A relative to universal set U is the set of all points which do not belongs to A and which, of course, belong to U.It is denoted by U – A or A’. 

We shall adopt the last notation, that is , A’ for the complement of A . 

    Thus  A’ = { x: x ∈U and x∈ A } 

In the future , by a complement of a set, we shall understand the complement of the set relative to the universal set unless otherwise stated. 

Note

                  (A’)’ = A 

De’ Morgan’s Law 

If A ,B and C are three sets , then  

1. (A∩ B )’ = A’∪ B’ 

1. (A ∪B )’ = A’∩ B’ 

Also , we have some additional results as

    (1)A – (B ∩C ) = (A -B ) ∪ ( A – C ) 

    (2)A – (B ∪C ) = ( A – B ) ∩ ( A – C ) 

Some Important Results on Number Of Elements in Sets 

   (1)n(A’) = n(U) – n(A) 

   (2)n(A∩ B’) = n(A) – n(A ∩B ) 

   (3)n(A∪ B ) = n(A) + n(B) – n(A∩ B ) 

   (4)n(A ∪B∪ C ) = n(A) + n(B) + n(C) – n( A∩ B ) – n(B∩ C ) – n(C∩ A ) + n(A ∩B∩ C ) 

Solved Examples

  (1)If U = { 1,2,3,4,5 } , A = { 4,5 } , then find A’ . 

 Sol.   A’ = U – A = { 1,2,3 } 

  (2)If A = { 1,2,3 } , B = { 2,3,4 } , C = {3,4,5,6 } and U = { 1,2,3,4,5,6,7 }, then find A’ , B’  and C’ . 

Sol.      A’ = { 4,5,6,7 }  

       B’ = { 1,5,6,7 }  

       C’ = { 1,2,7 }  

  (3)If U = { a, b ,c ,d , e ,f } and C =∅ , then find C’ . 

Sol.    C’ = U – C = ∅ 

  (4)If A = { 2,4,6,8 } , B = { 2,3,5,7} and U = { 1,2,3,4,5,6,7,8,9 } , then find  

             (A)(A∪ B )’       (B)(A∩ B )’ 

Sol.  

         (A) (A ∪B ) ‘ = A’ ∩B’ 

                             = { 1,3,5,7,9 }∩ { 1 ,4,6,8,9 } 

                             = { 1,9 } 

         (B)(A∩ B )’ = A’ ∪B’ 

                 = { 1,3,5,7,9 }∪ { 1,4,6,8,9 } 

                 = { 1,3,4,5,6,7,8,9,} 

   (5)If n(A) = 12 , n(B) = 8 , n(A ∩B ) = 4 , then find n(A∪ B ) . 

Sol.     We have ,  

                              n(A∪ B ) = n(A) + n(B) – n(A ∩B ) 

                                               = 12+8-4 = 16 

Cartesian Product of Two Sets 

  If a∈ A and b∈ B where A and B are set, then (a, b ) denotes what may be called an ordered pair whose first member is a and second member is b . Two ordered pairs (a, b ) and ( c, d ) are said to be equal if and only if  

                         a = c and b = d . 

* Thus the ordered pairs (1,2) and (2,1) are different. 

* The Cartesian Product of two sets A and B is the set { (a, b ) : a∈ A , b∈ B } and is denoted by A × B . 

* That is , the Cartesian product of two sets A and B is the set of all ordered pairs whose first member ( or first coordinate ) belongs to A and whose second member (or second coordinate ) belongs to B . 

Let A={x,y,z} and B={1,2,3}

Then A×B is calculated as shown below