Define the relation | on N\{0, 1} by a | b precisely when a divides evenly into b.   Identify all maximum, minimum, maximal, and minimal elements of |. Prove your result and if no maximum, minimum, maximal, and minimal elements exist explain why.

Let G = (V,E) be a bipartite graph, but this time it is a weighted graph. The weight of acomplete matching is the sum of the weights of its edges. We are interested in finding aminimum-weight complete matching in G.a) Give a legitimate C for a branch-and-bound (B&B) algorithm that finds a minimum-weightcomplete matching in G, and prove that your C is valid. Your C cannot be just the cost sofar.b) Using your C, apply B&B to find a minimum-weight complete matching in the followingweighted bipartite graph G: A = (1,2,3), B = (4,5,6,7)E = (((1,4), 3]. [(1,5), 4]. [(1,7). 15]. ((2,4), 1]. |(2,5), 8]. [(2,6), 3]. |(3,4), 3]. [(3,5), 9]. [ (3,6). 51).Show the solution tree. the C of every tree node generated, and the optimal solution. Also,mark the order in which each node in the solution tree is visited.

: Prove that for any n green and n red points in a plane such thatno 3 points are on the same line, it is possible to match eachgreen with one red such that no two segments connectingmatched nodes intersect (type the answer please)

Problem 5: Disjoint sets. Consider a list of cities c1, c2, . . . , cn. Assume we have a relationR such that, for any i, j, R(ci, cj ) is 1 if cities ci and cj are in the same province, and 0otherwise.(i) Using a disjoint set data structure, write an algorithm that puts each city in a setsuch that ci and cj are in the same set if and only if they are in the same province.Justify corectness of your algorithm.(ii) If a disjoint set data structure is implemented by trees and in union the name of thenew set is always the name of the larger set, what is the worst-case running time ofyour algorithm.

Let G := (V ,EG) be a 4-regular tree, i.e. an infinite connected undirected graph without cycles and such that each vertex has exactly 4 neighbors, the picture is a finite part of it. Recall that Th(G) denotes the theory of G, i.e. the set of all sentences that G satisfies. Using the Compactness theorem, prove that Th(G) has a disconnected model.  Hint: Add two new constant symbols and sneakily make them farther and farther from each other.

Let A = {1, 2, 3},         B = {{1}, 2, {1, 3}},         C = {1, {1}, 3}.   Compute the following sets:                                                                         2A∩C − B,         (A ∩ B) × (A − C),         (B ∪ C) − A.

For each of the following subsets of {0, 1}∗ tell whether or not it is regular. Proveyour answer.(a) {x | #1(x) = 2.#0(x)}(b) {x | #1(x) − #0(x) < 10}(c) {x | #1(x).#0(x) is even}

On S = f2; 4; 6; 10; 12; 20; 30; 60g we work with the divisibility relation R:xRy , x divides y:(i) Draw the Hasse diagram of R.(ii) Find all R-minimal elements.(iii) Find all R-maximal elements.(iv) Find all non-empty subsets U  S, so that 6 is an R-lowerbound of U.(v) Let A = f6; 10; 12g. If A admits an R-supremum,  nd it. If A admits an R-in mum,  nd it.(vi) Is R a lattice?