A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 2.4, Problem 11E
(a)
To determine
To show: there can be more than one top player.
(b)
To determine
To show: every n -player tournament has a top player.
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7. [10 marks]
Let G
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(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
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Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
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Find both a matching of maximum size and a vertex cover of minimum size in
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Chapter 2 Solutions
A Transition to Advanced Mathematics
Ch. 2.1 - The Cayley tables for operations o,*,+, and are...Ch. 2.1 - Let m,n and M=A:A is an mn matrix with real number...Ch. 2.1 - Let be an associative operation on nonempty set A...Ch. 2.1 - Let be an associative operation on nonempty set A...Ch. 2.1 - Suppose that (A,*) is an algebraic system and * is...Ch. 2.1 - Let (A,o) be an algebra structure. An element lA...Ch. 2.1 - Let G be a group. Prove that if a2=e for all aG,...Ch. 2.1 - Give an example of an algebraic structure of order...Ch. 2.1 - Prob. 9ECh. 2.1 - Construct the operation table for each of the...
Ch. 2.1 - Prob. 11ECh. 2.1 - Prob. 12ECh. 2.1 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 2.1 - Let m and a be natural numbers with am. Complete...Ch. 2.1 - Complete the proof of Theorem 6.1.4. First, show...Ch. 2.1 - Prob. 16ECh. 2.1 - Prob. 17ECh. 2.1 - Prob. 18ECh. 2.1 - Repeat Exercise 2 with the operation * given by...Ch. 2.2 - Prob. 1ECh. 2.2 - Let G be a group and aiG for all n. Prove that...Ch. 2.2 - Prove part (d) of Theorem 6.2.3. That is, prove...Ch. 2.2 - Prove part (b) of Theorem 6.2.4.Ch. 2.2 - List all generators of each cyclic group in...Ch. 2.2 - Let G be a group with identity e. Let aG. Prove...Ch. 2.2 - Let G be a group, and let H be a subgroup of G....Ch. 2.2 - Let ({0},) be the group of nonzero complex numbers...Ch. 2.2 - Prove that for every natural number m greater than...Ch. 2.2 - Show that the structure ({1},), with operation ...Ch. 2.2 - (a)In the group G of Exercise 2, find x such that...Ch. 2.2 - Show that (,), with operation # defined by...Ch. 2.2 - Prob. 13ECh. 2.2 - Prob. 14ECh. 2.2 - Prob. 15ECh. 2.2 - Show that each of the following algebraic...Ch. 2.2 - Prob. 17ECh. 2.2 - Given that G={e,u,v,w} is a group of order 4 with...Ch. 2.2 - Give an example of an algebraic system (G,o) that...Ch. 2.2 - (a)What is the order of S4, the symmetric group on...Ch. 2.3 - Find the order of the element 3 in each group....Ch. 2.3 - Find the order of each element of the group S3....Ch. 2.3 - Let 3 and 6 be the sets of integer multiples of 3...Ch. 2.3 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 2.3 - Let ({a,b,c},o) be the group with the operation...Ch. 2.3 - (a)Prove that the function f:1824 given by f(x)=4x...Ch. 2.3 - Define f:1512 by f(x)=4x. Prove that f is a...Ch. 2.3 - Let (G,) and (H,*) be groups, i be the identity...Ch. 2.3 - Show that (4,+) and ({1,1,i,i},) are isomorphic.Ch. 2.3 - Prove that every subgroup of a cyclic group is...Ch. 2.3 - Let G=a be a cyclic group of order 30. What is the...Ch. 2.3 - Assign a grade of A (correct), C (partially...Ch. 2.3 - Find all subgroups of (8,+). (U11,). (5,+). (U7,)....Ch. 2.3 - In the group S4, find two different subgroups that...Ch. 2.3 - Prove that if G is a group and H is a subgroup of...Ch. 2.3 - (a)Prove that if H and K are subgroups of a group...Ch. 2.3 - Let G be a group and H be a subgroup of G. If H is...Ch. 2.3 - Prove or disprove: Every abelian group is cyclic.Ch. 2.3 - Let G be a group. If H is a subgroup of G and K is...Ch. 2.4 - Define f:++ by f(x)=x where + is the set of all...Ch. 2.4 - Assign a grade of A (correct), C (partially...Ch. 2.4 - Define f: by f(x)=x3. Is f:(,+)(,+) operation...Ch. 2.4 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 2.4 - Let f the set of all real-valued integrable...Ch. 2.4 - Prob. 6ECh. 2.4 - Let M be the set of all 22 matrices with real...Ch. 2.4 - Let Conj: be the conjugate mapping for complex...Ch. 2.4 - Prove the remaining parts of Theorem 6.4.1.Ch. 2.4 - Is S3 isomorphic to (6,+)? Explain.Ch. 2.4 - Prob. 11ECh. 2.4 - Use the method of proof of Cayley's Theorem to...Ch. 2.5 - Let (R,+,) be an algebraic structure such that...Ch. 2.5 - Assign a grade of A (correct), C (partially...Ch. 2.5 - Which of the following is a ring with the usual...Ch. 2.5 - Let [2] be the set {a+b2:a,b}. Define addition and...Ch. 2.5 - Complete the proof that for every m,(m+,) is a...Ch. 2.5 - Define addition and multiplication on the set ...Ch. 2.5 - Prob. 7ECh. 2.5 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 2.5 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 2.5 - Prob. 10ECh. 2.5 - Prob. 11ECh. 2.5 - Prob. 12ECh. 2.5 - Prob. 13ECh. 2.5 - Prob. 14ECh. 2.6 - Prob. 1ECh. 2.6 - Let A and B be subsets of . Prove that if sup(A)...Ch. 2.6 - (a)Give an example of sets A and B of real numbers...Ch. 2.6 - (a)Give an example of sets A and B of real numbers...Ch. 2.6 - Prob. 5ECh. 2.6 - Prob. 6ECh. 2.6 - Prob. 7ECh. 2.6 - Prob. 8ECh. 2.6 - Prob. 9ECh. 2.6 - Prob. 10ECh. 2.6 - Prob. 11ECh. 2.6 - Prob. 12ECh. 2.6 - Prob. 13ECh. 2.6 - Prob. 14ECh. 2.6 - Prob. 15ECh. 2.6 - Prob. 16ECh. 2.6 - Use the definition of “divides” to explain (a) why...Ch. 2.6 - Prob. 18ECh. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - For each function, find the value of f at 3 and...Ch. 2.6 - Let A be the set {1,2,3,4} and B={0,1,2,3}. Give a...Ch. 2.6 - Formulate and prove a characterization of greatest...Ch. 2.6 - Prob. 24ECh. 2.6 - Prob. 25E
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- 5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forwardQ/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forward
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