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4-Dimensional Matching is in NP, since we can check in O(n) time, using an n times 4 array initialized to all 0, that a given set of n 4-tuples is disjoint. We now show that 3-Dimensional Matching can be reduced to 4-Dimensional Matching in polynomial time. So consider an instance of 3-Dimensional Matching, with sets X, Y, and Z of size n each, and a collection C of ordered triples. We define an instance of 4-Dimensional Matching as follows. We have sets W, X, Y, and Z, each of size n, and a collection C' of 4-tuples defined so that for every (x_j, y_k, z_l) in C, and every i between 1 and n, there is ... See the full answer