We start by defining the Independent Set problem. Given a graph G-(V, E), we say a set of nodes S C V is independent if no two nodes in S are joined by an edge. The Independent Set problem, which we denote as IS, is the following. Given G, find an independent set that is as large as possible. Stated as a decision problem, IS answers the question, does there exist a set S E G such that |SI S k? Then set k as large as possible. For this problem, you may take as given that IS is NP-complete A store trying to analyze the behavior of its customers will often maintain a table A where the rows of the table correspond to the customers and the columns (or fields) corresponding to prod- ucts the store sells. The entry A[i,3] specifies the quantity of product J that has been purchased by customers i. For example, Table 1 shows one such table. Table 1: Customer Tracking Table Customer Detergent Beer Diap ers Cat Litter Raj 6 0 0 0 Chelsea 0 0 One thing that a store might want to do with this data is the following. Let's say that a subset S of the customers is diverse if no two of the customers in S have ever bought the same product (i.e., for each product, at most one of the customers S has ever bought it). A diverse set of customers can be useful, for example, s a target pool for market research. We can now define the Diverse Subset problem (DS) as follows: Given an mx n array A as defined above and a number k m, is there a subset of at least k customers that is divers? Prove that DS is N P-complete