Operations of Sets

(1 ) Union of two sets

      If A and B are two sets, then the set of all elements which either belongs to A or to   B is called the union of the two sets and is denoted by  

                   A∪ B

 It is usually read “A union B”. 

 Thus , A∪ B = { x: x ∈ A or x∈ B }. 

Properties of the Union of Sets

      If A , B and C are three Sets , then  

(1)Idempotent law

               A∪ A  = A 

(2)Commutative law

      A∪ B = B ∪A 

(3)Associative law

             A∪ ( B ∪C ) = (A ∪B ) ∪ C 

(4)Identity law

     (a) A∪∅   = A ,       (b) A ∪U = U 

     Where U is an universal set and ∅ is an empty set . 

Example 

1. If A = { a b ,c ,d } and B = { b , d ,e , f } , then find A ∪B . 

Sol.  A ∪B = { a , b ,c , d , e , f } 

(2)Intersection of Sets

      If A and B are given sets , then the intersection of A and B is the set of all elements   which belong to both A and B and is denoted by  

                              A∩ B  

It is usually read “A intersection B”.  

Thus ,   A ∩B = { x: x∈ A and x ∈B } . 

Properties of the Intersection of Sets

       If A, B, and C are three sets, then  

(1)Idempotent law

                                        A∩ A = A  

(2)Commutative law 

                                         A ∩B = B∩ A  

(3)Associative law

                                    A ∩ (B ∩C ) = (A∩ B ) ∩ C 

(4)Identity law

                              (a)  A ∩U = A ,    (b) A ∩ ∅= ∅ 

(5)Distributive law

                            (a)    A∩ (B ∪C ) = (A∩ B ) ∪ ( A ∩C ) 

                            (b)    A ∪ ( B ∩C ) = ( A∪ B ) ∩  ( A ∪C )  

Solved Examples

       (1)If A = { 1,2, 3,4,6,12 } , B = { 2,4,6,8,10 } , then find A∩ B . 

Sol.       A ∩B = { 2,4,6 } 

(3) Difference of two Sets 

       If A and B are two sets, then the difference of A and B is defined to be the set  

                   A – B = { x : x∈  A , x ∉B }  

Similarly ,   B – A = { x: x ∈B , x ∉A } 

Solved Examples

     (1)If A = { 1,2,3,4,5 } , B = { 2,4,5,6,7 } , then find A – B. 

Sol.        A – B = { 1,3 } 

    (2)If A = { 1,3,5,7 } , B = { 2,6,8,10 } , then find A – B and B – A . 

Sol.         A – B = { 1,3,5,7 }  

               B – A = { 2,6,8,10 }  

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

Sol.     A ∪B∪ C = { 1,2,3,4,5,6 } 

    (4)If A = { x: x² - 2x +1 =0 } , B = { x: x³ – x² - 2x = 0 } , then find A ∩B . 

Sol.             A = { x: x² -2x +1 = 0 }  

                       = { x: (x -1)(x-1) = 0 } 

                       = { 1 } 

                  B = { x :x³ – x² - 2x = 0 }  

                      = { x : x (x +1 )(x – 2) = 0 }  

                      = { -1 , 0 , 2 }  

Therefore ,   A ∩B = ∅ 

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

Sol.       A – A = ∅ .