Define a relation S on B={a,b,c,d} by S={(a,b),(a,c),(b,c),(d,d)}. Draw the directed graph of relation S.

(8) Consider an undirected graph G, and the relation R(u, v) that is defined as "there is a path from u to v". Which of the following statements are true for R? [O Ris reflexive. [] Ris transitive. [ ] Ris symmetric. [ ] Ris asymmetric. [ ] Ris antisymmetric. [ ] Ris an equivalence relation. [] Ris a partial order. [] Ris a total order. (9) If you perform a breadth-first search (BFS) on the given graph, starting from node 6, which of the edges of the graph will be part of the BFS tree? Assume that ties are broken in favor of lower-numbered nodes. [ ] (6, 4) [] (6,5) [ ] (4, 3) [ ] (4, 2) [] (5, 4) [] (5, 2) [O (5, 1) [ ] (2,1)

Let A be the towns {P, Q, R, S}. Let B be the one way street relation {(P, S), (Q, P), (R, P), (S, Q), (S, R)}. Find B^2 (pairs of towns that are connected by two streets). Find B3 (pairs of towns that are connected by three streets). You must use directed graph to find the answer.

Adjacency matrix. Given a relation relation (as a numpy.array, see example below) and number of nodes n, define a function adjacency to construct the adjacency matrix A (as a numpy.array).You can assume that n is a positive integer and treat relation as a list of directed edges.   # Example from p112 of VMLSrelation = np.array([[1, 2], [1, 3], [2, 1], [2, 4], [3, 4], [4, 1]])n = 4 # Modify the function adjacency below with your answer for Q8def adjacency(relation, n):    A = None    ###########################    ### YOUR CODE HERE ########            ## REMOVE THIS LINE--------    raise NotImplementedError()    ## ------------------------    ###########################    return A

The USES relation naturally forms a graph. As mentioned in the lecture, the USES relation graph should be a Directed Acyclic Graph (DAG). (1) A graph could be represented with two different data structures, i.e., adjacency list and adjacency matrix. Represent the USES relation graph in Figure 3-1 with adjacency list and adjacency matrix. 3 2 4 5 6. Figure 3-1: USES relation graph of the modules in a system (2) Suppose the input of a USES relation graph is represented with adjacency list, develop a function in pseudocode to check whether the USES relation graph is an acyclic graph, using Stepwise Refinement technique taught in lectures. You should provide with at least 2 refinements. Assuming there are V vertices and E edges, please state the time complexity of your algorithm.

1. Let A = {0, 1,2,3} and let R be a relation on A such that Ry if and only if (a) List the elements of R. (b) Draw a directed graph representation of R.. y is odd. (c) Is R reflexive? Is R symmetric? Is R transitive? Justify your answer to each.

True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.

Discrete Math

5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…

Can you help me solve this exercise? Please note that the greedy approach described in the advice paragraph does not work.