Exercise $2\mathrm{.}3\mathrm{.}3$: In the form of relational algebra implemented in SQL$,$ relations are not sets, but bags; that is$,$ tuples are allowed to appear more than once. There are extended definitions of union, intersection, and difference for

bags, which we shall define below. Write MapReduce algorithms for computing

the following operations on bags R and S:

$($a$)$ Bag Union, defined to be the bag of tuples in which tuple t appears the

sum of the numbers of times it appears in R and S$.$

$($b$)$ Bag Intersection, defined to be the bag of tuples in which tuple t appears

the minimum of the numbers of times it appears in R and S$.$

$($c$)$ Bag Difference, defined to be the bag of tuples in which the number of

times a tuple t appears is equal to the number of times it appears in R

minus the number of times it appears in S$.$ A tuple that appears more

times in S than in R does not appear in the difference.