1.3)    In each case, find the simpler expression representing the same SET.
              Assume A and B are sets.

    Solution:

            Easiest way to make one understand the concepts is by Venn Diagrams.
            Some of the problems of this excercise are solved below using Venn Diagrams.

            (a)  A - (A - B)
                  Let us start out by drawing a Venn for A - B

                                      A - B  is represented by the yellow region.

            Set A is shown below

            From A we have to take of A - B, the Venn is shown below

                                                         A - (A - B)

            If one can carefully observer the area covered by A - (A - B)  one can easily recognize
            that it is same as the A  B region.
            So we can use A  B in place of A - (A - B)

                               A - (A - B) = A  B

                                       Or

            Alternate Way to prove A - (A - B) = A   B is illustrated below:

            Here we use the set identities described earlier and not the Venn Diagrams. This becomes
            necessary in case you have more than 3 sets, as it will be very complicated if would try to
            illustrate. This method is always preferrable as it doesnot have any limitation on number of
            sets involved.

            A - B = A  B' has to be kept in mind all the time as this helps out to solve many problems.
            One can easily prove the above using the Venn Diagrams as is left to students as an excercise.

            LHS = A - (A - B)
                    = A - ( A  B')                   // a - b = a  b' 
                    now consider A  B' as B.
                    then we can rewrite the above expression as
                    = A  ( A  B')'
                    = A  ( A'  B)               // using the identity( A  B)' = A'   B'
                    = (A  A')  (A  B)
                    =  (A  B)                         // using the identity  ( A  A') = Empty Set
 
                    So,  A - (A - B) = (A  B)

                In the same way one can get the simpler expression of each of the excercise problems
                and one  can also develop his/her manipulating standards. ( One can improve his analytical
                skills)

           (d) (A - B)  (B - A)  (A  B)

              Starting out with Venn diagrams for A -B, B - A and A  B.

                                                                            A  B
                (A - B)  (B - A)  (A  B) is the area covered by anyone of  A - B or B -A or
                A  B.
                Venn Diagram of (A - B)  (B - A)  (A  B) is shown below

                From the above Venn Diagram it is pretty clear that the area covered by
                (A - B) (B - A) (A  B) is same as the A  B.
                We can replace (A - B) (B - A)  (A  B)  with A  B.

                           (A - B) (B - A)  (A  B) = A  B
 

        ( e )    (A'  B')'

                  Let us start out with A' B' and at last the (A'  B')'.
                  A'  B' are shown below.

                  The green color represents the      A' B'  area.

                  The complement of A' B' is the white area between the green area in the above
                  Venn diagram.

                The above area if observed carefully is also the same for A  B.
                So we can conclude as      .

                                       (A'  B')' =  A  B