please show the induction proof

4. [30 points] Collaborative Problem: We start by defining the Independent Set Problem (IS). Given a graph G = (V, E), we say a set of nodes S CV is independent if no two nodes in S are joined by an edge. The Independent Set Problem, which we denote 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 CV such that |SI > 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) correspond to products the store sells. The entry A[i,j] specifies the quantity of product that has been purchased by customer i. For example, Table ?? shows one such table. Table 1: Customer Tracking Table Customer Detergent Beer Diapers Cat Litter Raj 0 6 0 3 Alanis 0 Chelsea 2 0 3 0 0 7 (Continued on next page) 1 One thing that a store might want to do with this data is the following. Let's say that a subset 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 in S has ever bought it). A diverse set of customers can be useful, for example, as 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