Programming with Lists Multisets, or bags, can be represented as list of pairs (x, n) where n indicates the number of occurrences of x in the multiset. type Bag a - [(a, Int)] For the following exercises you can assume the following properties of the bag representation. But note: Your function definitions have to maintain these properties for any multiset they produce! (1) Each element x occurs in at most one pair in the list. (2) Each element that occurs in a pair has a positive counter. As an example consider the multiset {2, 3, 3,5,7,7,7,8), which has the following representation (among others) Note that the order of elements is not fixed. In particular, we cannot assume that the elements are sorted. Thus the above list representation is just one example of several possible. (a) Define the function ins that inserts an element into a multiset. ins :: Eq a => a-> Bag a-> Bag a (Note: The class constraint "Eq a =>" restricts the element type a to those types that allow the comparison of elements for equality with --.) (b) Define the function del that removes an element from a multiset. del Eq a -> Bag a Bag a (c) Define a function bag that takes a list of values and produces a multiset representation bag :: Eq a => [a] -> Bag a For example, with xs7,3,8,7,3,2,7,5] we get the following result. > bag xs (Note: It's a good idea to use of the function ins defined earlier.) (d) Define a function subbag that determines whether or not its first argument bag is contained in the second. subbag Eq aBag a Bag aBool Note that a bag b is contained in a bag b' if every element that occurs n times in b occurs also at least n times in b'. (e) Define a function isbag that computes the intersection of two multisets. isbag Eq Bag a ->Bag a -Bag a (0) Define a function size that computes the number of elements contained in a bag. sizeBag a ->Int