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Questions and Answers of
Linear Algebra
Let T = (V, E) be a rooted tree with root r. Define the relation R on V by x R y, for x, y ∈ V, if x = y or if x is on the path from r to y. Prove that R is a partial order.
Let T = (V, E) be a tree with V = {v1, v2, ..., vn}, for n ¥ 2. Prove that the number of pendant vertices in T is equal to
Let G = (V, E) be a loop-free undirected graph. Define the relation R on E as follows: If e1, e2 ∈ E, then e1 R e2 if e1 = e2 or if e1 and e2 are edges of a cycle C in G. (a) Verify that R is an
If G = (V, E) is a loop-free connected undirected graph and a, b V, then we define the distance from a to b (or from b to a), denoted d(a, b), as the length of a shortest path (in G)
Let G = (V, E) be a weighted graph, where for each edge e = (a, b) in E, wt(a, b) equals the distance from a to b along edge e. If (a, b) ∉ E, then wt(a, b) = ∞.Fix v0 ∈ V and let S ⊂ V, with
(a) Apply Dijkstra's algorithm to the weighted graph G = (V, E) in Fig. 13.4, and determine the shortest distance from vertex a to each of the other six vertices in G. Here wt(e) = wt(x, y) = wt(y,
(a) Apply Dijkstra's algorithm to the graph shown in Fig. 13.1 and determine the shortest distance from vertex a to each of the other vertices in the graph.(b) Find a shortest path from vertex a to
Use the ideas developed at the end of the section to confirm the result obtained in (a) Example 13.2; and (b) part (a) of Exercise 2.
Prove or disprove the following for a weighted graph G = (V, E), where V = {v0, v1, v2, .. . , vn] and e1 ∈ E with wt(e1) < wt(e) for all e ∈ E, e ≠ e1. If Dijkstra's algorithm is applied to G,
Apply Kruskal's and Prim's algorithms to determine minimal spanning trees for the graph shown in Fig. 13.8.
Let G = W4, the wheel on four spokes. Assign the weights 1, 1, 2, 2, 3, 3, 4, 4 to the edges of G so that (a) G has a unique minimal spanning tree; (b) G has more than one minimal spanning tree.
Let G = (V, E) be a loop-free weighted connected undirected graph with T = (V, E'), a minimal spanning tree for G. For v, w ∈ V, is the path from v to w in T a path of minimum weight in G?
Table 13.1 provides information on the distance (in miles) between pairs of cities in the state of Indiana.A system of highways connecting these seven cities is to be constructed. Determine which
(a) Answer Exercise 4 under the additional requirement that the system includes a highway directly linking Evansville and Indianapolis.(b) If there must be a direct link between Fort Wayne and Gary
Let G = (V, E) be a loop-free weighted connected undirected graph. For n ∈ Z+, let [e1, e2, . . . , en] be a set of edges (from E) that includes no cycle in G. Modify Kruskal's algorithm in order
(a) Modify Kruskal's algorithm to determine an optimal tree of maximal weight.(b) Interpret the information of Exercise 4 in terms of the number of calls that can be placed between pairs of cities
Prove Theorem 13.2.
(a) For the network shown in Fig. 13.20, let the capacity of each edge be 10. If each edge e in the figure is labeled by a function f, as shown, determine the values of s,t,w,x, and y so that f is a
Prove Corollaries 13.3 and 13.4.
Find a maximum flow and the corresponding minimum cut for each transport network shown in Fig. 13.21.
Apply the Edmonds-Karp and Ford-Fulkerson algorithms to find a maximum flow in Examples 13.12, 13.13, and 13.14.
Prove Corollary 13.5.
In each of the following "transport networks" two companies, ci and C2, produce a certain product that is used by two manufacturers, mi and m2. For the network shown in part (a) of Fig. 13.22,
Find a maximum flow for the network shown in Fig. 13.23. The capacities on the undirected edges indicate that the capacity is the same in either direction. [However, for an undirected edge a flow can
Let A1, A2, . . ., An be a collection of sets, where A1 = A2 = ..... = An and | At | = k > 0 for all 1 < i < n. (a) Prove that the given collection has a system of distinct representatives
Let G = (V, E) be a bipartite graph, where V is partitioned as X ∪ Y. If deg(x) > 4 for all x ∈ X and deg(y) < 5 for all y ∈ F, prove that if |X| < 10 then 8(G) < 2.
Let G = (V, E) be bipartite with V partitioned as X ∪ Y. For all x ∈ X, deg(x) > 3, and for all y ∈ Y, deg(y) < 7. If |X| < 50, find an upper bound (that is as small as possible) on δ(G).
(a) Let G = (V, E) be the bipartite graph shown in Fig. 13.32, with V partitioned as X ˆª Y. Determine 8(G) and a maximal matching of X into Y.b) For any bipartite graph G = (V, E), with V
For n > 2, prove that the hypercube Qn has at least 2(2n- 2) tion 11.5.) perfect matchings (as defined above in Exercise 5).
Cathy is liked by Albert, Joseph, and Robert; Janice by Joseph and Dennis; Theresa by Albert and Joseph; Nettie by Dennis, Joseph, and Frank; and Karen by Albert, Joseph, and Robert, (a) Set up a
At Rydell High School the senior class is represented on six school committees by Annemarie (A), Gary (G), Jill (J), Kenneth (K), Michael (M), Norma (N), Paul (P), and Rosemary (R). The senior
Let G = (V, E) be a bipartite graph with V partitioned as X ∪ Y, where X = {x1, x2, . . ., xm] and Y = {x1, x2, . . . , xn}- How many complete matchings of X into Y are there if (a) m = 2, n = 4,
If G = (V, E) is an undirected graph, a spanning subgraph H of G in which each vertex has degree 1 is called a one-factor (or perfect matching) for G.a) If G has a one-factor, prove that |V| is
Prove Corollary 13.6.
Fritz is in charge of assigning students to part-time jobs at the college where he works. He has 25 student applications, and there are 25 different part-time jobs available on the campus. Each
For each of the following collections of sets, determine, if possible, a system of distinct representatives. If no such system exists, explain why. (a) A1 = {2, 3, 4}, A2 = {3, 4}, A3 = {1}, A4 = {2,
(a) Determine all systems of distinct representatives for the collection of sets A1 = {1, 2}, A2 = {2, 3}, A3 = {3, 4}, A4 = {4, 1}. b) Given the collection of sets A1 = {1, 2}, A2 = {2, 3}, . . . ,
Apply Dijkstra's algorithm to the weighted directed multigraph shown in Fig. 13.33, and find the shortest distance from vertex a to the other seven vertices in the graph.
For her class in the analysis of algorithms, Stacy writes the following algorithm to determine the shortest distance from a vertex a to a vertex b in a weighted directed graph G = (V, E). Step 1 :
(a) Let G = (V, E) be a loop-free weighted connected undirected graph. If e1 ∈ E with wt(ei) < wt(e) for all other edges e1 ∈ E, prove that edge e1 is part of every minimal spanning tree for
(a) LetG = (V, E) be a loop-free weighted connected undirected graph where each edge e of G is part of a cycle. Prove that if e1 ∈ E with wt(e1) > wt(e) for all other edges e e E, then no
Using the concept of flow in a transport network, construct a directed multigraph G = (V, E), with V = {u, v, w, x, y} and id(u) = 1, od(u) = 3; id(v) = 3, od(v) = 3; id(w) = 3, od(w) = 4; id(x) = 5,
A set of words {qs, tq, ut, pqr, srt] is to be transmitted using a binary code for each letter, (a) Show that it is possible to select one letter from each word as a system of distinct
This exercise outlines a proof of the Birkhoff-von Neumann Theorem.(a) For n ˆˆ Z+, an n × n matrix is called a permutation matrix if there is exactly one 1 in each row and column, and all
Find the additive inverse for each element in the rings of Examples 14.5 and 14.6.
Let (Q, ⊕, ⊙) denote the field where 0 and O are defined byA ⊕ b = a + b - k, a ⊙ b = a + b + (ab/m),for fixed elements k, m (≠ 0) of Q. Determine the value for k and the value for m in
Let R = {a + bi|a, b ∈ Z, i2 = -1}, with addition and multiplication defined by (a + bi) + (c + di) = (a + c) + (b + d)i and (a + bi)(c + di) = (ac - bd) + (be + ad)i, respectively. (a) Verify that
a. Determine the multiplicative inverse of the matrixin the ring ^{2)- that is, find a, b, c, d so that b. Show that s a unit in the ring M2(Q) but not a unit in M2 (Z).
Ifis a unit of this ring if and only if ad - be 0.
Give an example of a ring with eight elements. How about one with 16 elements? Generalize.
For R = {s, t, x, y}, define + and , making R into a ring, by Table 14.5(a) for + and by the partial table for + in Table 14.5(b).(a) Using the associative and distributive laws, determine the
Determine whether or not each of the following sets of numbers is a ring under ordinary addition and multiplication. a) R = the set of positive integers and zero b) R = {kn|n ∈ Z, k a fixed
Let (R, 4-, •) be a ring with a, b, c, d elements of R. State the conditions (from the definition of a ring) that are needed to prove each of the following results. a) (a + b) + c) = b + (c +a) b)
Consider the set Z together with the binary operations ⊕ and ⊙ given in Example 14.3. (a) Verify the associative laws for ⊕ and ⊙ and the distributive laws in order to complete the work
Define the binary operations ⊕ and ⊙ on Z by x ⊕ y = x + y - 7,x ⊙ y = x + y - 3xy, for all x, y ∈ Z. Explain why (Z, ⊕, ⊙) is not a ring
Let k, m be fixed integers. Find all values for k, m for which (Z, ⊕, ⊙) is a ring under the binary operations x ⊕ y = x + y - k, x ⊙ y = x + y - mxy, where x, y ∈ Z.
Tables 14.4(a) and (b) make (R, +, •) into a ring, where R = {s, t, x, y}. (a) What is the zero for this ring? (b) What is the additive inverse of each element? (c) What is t(s + xy)? (d) Is R
Define addition and multiplication, denoted by Š• and Š™, respectively, on the set Q as follows. For a, b ˆˆ Q, a Š• b=a + b + 7, a Š™ b = a + b + (ab/1). (a) Prove that (Q, Š•,
Complete the proofs of Theorems 14.2, 14.4, 14.5, and 14.10.
(a) Let (R, +, •) be a finite commutative ring with unity u. If r ∈ R and r is not the zero element of R, prove that r is either a unit or a proper divisor of zero.(b) Does the result in part (a)
(a) For R = M2(Z), prove thatis a subring of R. (b) What is the unity of R? (c) Does S have a unjty ? (d) Does S have any properties that R does not have? (e) Is S an ideal of R?
Let S and T be the following subsets of the ring R = M2 (Z) :(a) Verify that S is a subring of R. Is it an ideal?(b) Verify that T is a subring of R. Is it an ideal?
Let (R, +, •) be a commutative ring, and let z denote the zero element of R. For a fixed element a ∈ R, define N(a) = [r ∈ R | ra = z}. Prove that N(a) is an ideal of R.
Let R be a commutative ring with unity u, and let I be an ideal of R. (a) If u ∈ 1, prove that I = R. (b) If I contains a unit of R, prove that I = R.
Let (R, +, €¢) be the (finite) commutative ring with unity given by Tables 14.6(a) and (b).(a) Verify that R is a field.(b) Find a subring of R that is not an ideal.(c) Let x and y be unknowns.
Let R be a commutative ring with unity u.a) For any (fixed) a ∈ R, prove that aR = {ar| r ∈ R] is an ideal of R.b) If the only ideals of R are {z} and R, prove that R is a field.
Let (S, +, •) and (T, +', •') be two rings. For R = S × T, define addition "⊕" and multiplication "⊙" by(s1, t1) ⊕ (s2, t2) = (S1 + S2, t1 +' t2),(s1 t1) ⊙ (s2, t2) = (S1 • s2, t1.
Let (R, +, •) be a ring with unity u, and |R| = 8. On R4 = R × R × R × R, define + and • as suggested by Exercise 18. In the ring R4, (a) How many elements have exactly two nonzero
If a, b, and c are any elements in a ring (R, +, •), prove that (a) a(b - c) = ab - (ac) = ab - ac and (b) (b - c)a = ba - (ca) = ba - ca.
Let (R, +, •) be a ring, with a ∈ R. Define 0a = z, la = a, and (n + 1)a = na + a, for all n ∈ Z+. (Here we are multiplying elements of R by elements of Z, so we have yet another operation that
(a) For ring (R, +, •) and each d e R, we define al = a, and dn+l = and, for all n ∈ Z+. Prove that for all m, n ∈ Z+, (am)(an) = am+n and (am)n = amn.(b) Can you suggest how we might define
a. If R is a ring with unity and a, b are units of R, prove that ab is a unit of R and that (ab)-l = b-la-l.b. For the ring M2(Z), find A-1 B-1 (AB)-1 (BA)-1and B-1A-1 if
Prove that a unit in a ring R cannot be a proper divisor of zero.
(a) Verify that the subsets S = {s, w] and T = {s, v, x] are subrings of the ring R in Example 14.6. (The binary operations for the elements of S, T are those given in Table 14.3.)b) Are the subrings
Let S and T be subrings of a ring R. Prove that S ∩ T is a subring of R.
Let R = M2(Z) and let S be the subset of R whereProve that S is a subring of R.
Let (R, +, •) be a ring. If S, T1 and T2 are subrings of R, and S ⊂ T1 ∪ T2, prove that S ⊂ T1 or S ⊂ T2.
(a) Determine whether each of the following pairs of integers is congruent modulo 8.(i) 62,118 (ii) -43,-237 (iii) -90, 230(b) Determine whether each of the following pairs of integers is congruent
Complete the proofs of Theorems 14.11 and 14.12.
Define relation R on Z+ by a R b, if r (a) = x(b), where x (a) = the number of positive (integer) divisors of a. For example, 2 R 3 and 4 R 25 but 5R9.a) Verify that 2ft is an equivalence relation on
Find the multiplicative inverse of each element in Z11, Z13, and Z17.
Find [a]-1 in Z1009 for (a) a = 17, (b) a = 100, and (c) a = 111
(a) Find all subrings of Z12, Z18, and Z24. (b) Construct the Hasse diagram for each of these collections of subrings, where the partial order arises from set inclusion. Compare these diagrams with
How many units and how many (proper) zero divisors are there in (a) Z17 (b) Zn117? (c) Z1117?
Prove that in any list of n consecutive integers, one of the integers is divisible by n.
If three distinct integers are randomly selected from the set {1, 2, 3, . . . , 1000}, what is the probability that their sum is divisible by 3?
(a) For c, d, n, m ∈ Z, with n > 1 and m > 0, prove that if c = d (mod n), then me = md (mod n) and cm = dm (mod n).b) If xnxn-1 ∙∙∙∙∙∙∙ x1\x0 = xn • 10 + ∙∙∙∙∙∙ +
(a) Prove that for all n ∈ N, 10" = (-1)" (mod 11).(b) Consider the result for mod 9 in part (b) of Exercise 18. State and prove a comparable result for mod 11.
For each of the following determine the value(s) of the integer n > 1 for which the given congruence is true.(a) 28 = 6 (mod n) (b) 68 = 37 (mod n)(c) 301 ee 233 (mod n) (d) 49 = 2 (mod n)
For p a prime determine all elements a ∈ Zp where a2 = a.
For a, b, n ∈ Z+ and n > 1, prove that a = b (mod n) => gcd(a, n) = ged(b, n).
(a) Show that for all [a] ∈ Z7, if [a] ≠ [0], then[a]6 = [1],(b) Let n ≠ Z+ with ged(n, 7) = 1. Prove that 7|(n6 - 1).
Use the Caesar cipher to encrypt the plaintext: "All Gaul is divided into three parts."
The ciphertext FTQIMKIQIQDQ was encrypted using the encryption function E: Z26 → Z26 where E(θ) = (θ + k) mod 26. Considering the frequencies of occurrence for the letters in the ciphertext,
Determine the total number of affine ciphers for an alphabet of (a) 24 letters; (b) 25 letters; (c) 27 letters; and (d) 30 letters.
The ciphertext RWJWQTOOMYHKUXGOEMYP was encrypted with an affine cipher. Given that the plaintext letters e, t are encrypted as the ciphertext letters W, X, respectively, determine (a) the encryption
(a) How many distinct terms does the linear congruential generator with a = 5, c = 3, m = 19, and x0 = 10, produce? (b) What is the sequence of pseudorandom members generated?
Given the modulus m and the two seeds x0, x1, with 0 < x0,x1 < m, a sequence of pseudorandom numbers can be generated recursively from xn = (xn-1 + xn-2) mod m, n > 2. This generator is called the
Let xn+1 = (axn + c) mod m, where 2 < a < m, 0 < c < m, 0 < x0 < m, 0 < xn+i < m, and n > 0. Prove that xn = (anx0 + c[(an - 1)/(a - 1)]) mod m, 0 < xn < m.
Consider the linear congruential generator with a = 7, c = 4, and m = 9. If x4 = 1, determine the seed x0.
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