3. Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e):w(e) + c for all e Є E(G).=(b) What is the analogous claim for shortest paths in (G, w) and (G,m)? Provethis claim or provide a counterexample showing that it is not true.

(b)The analogous claim for shortest paths in (G, w) and (G, m) is true. That is, if P is the…

Answered: Consider a network (G, w). Let c ER,…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 30E

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3. Consider a formula Væ3y3zVuE(x, y) ^ E(x, z) ^ (u = x V ¬E(u, y) V ¬E(u, z)). Here, 2, y, z, u e V for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n. (a) What is the maximal possible number of edges in an undirected graph satisfying this formula? (b) What is the minimal possible number of edges in an directed graph satisfying this formula? Show both that there exists a graph with that many edges (you can just describe how to construct such a graph), and that this number is optimal.
2. Suppose n ≥ 1 is an integer. Consider an (n + 1) x (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0, 0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or ‘up' (i.e. from (a, b) to (a, b+1)). (a) Draw all distinct Binomial Paths with 2n steps when n = = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B₁, be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn Sn. (You don't have to prove that it is a bijection.)
5. a) Determine the converse, inverse and contrapositive of the following statements by indicating the conditional statement. (i) If we are in binary system then 1 + 1 = 10 (ii) Good food are not cheap (ii) If 9x +36 = 9 then x # 17. (iv) If cos(x) =1 then x = 0. b) If f(x) = 8x, g(x)=x*and h(x) = 4x +3 then find (f°g) h and f(g h). Show that (f °g) h= f •(g•h)? c) Let f:R→(-1, 1) be defined by f(x) = -,xe R . Find the X-1 inverse of the above function if it exists, where R is the set of real numbers.
1. Decide if the following statement is True or False: Let A1, A2, ..., Ak E Mnxn(R) be such that Mnxn(R) = span{A1, A2,..., Ak}. Then Mnxn(R) = span{Af , A,..., A}. Note; AT denotes the transpose of A.
Let G = (V,E) be a connected graph, and u, v in V The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices. Show that if A(G) ≥ 4, then A(Ḡ) ≤ 2. Show that if G has a cut vertex and A(G) = 2, then Ḡ has a vertex without neighbors.

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Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e) = w(e)+c for all e € E(G). (b) What is the analogous claim for shortest paths in (G, w) and (G,m)? Prove this claim or provide a counterexample showing that it is not true.