Discrete Mathematics: Introduction to Mathematical Reasoning
1st Edition
ISBN: 9780495826170
Author: Susanna S. Epp
Publisher: Cengage Learning
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Chapter 4.5, Problem 27ES
To determine
Tofind:For all integers
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6. [10 marks]
Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
T.
(a) How many vertices does BL(T) have?
(b) How many edges does BL(T) have?
Prove that your answers are correct.
4. [10 marks]
Find both a matching of maximum size and a vertex cover of minimum size in
the following bipartite graph. Prove that your answer is correct.
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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).
Chapter 4 Solutions
Discrete Mathematics: Introduction to Mathematical Reasoning
Ch. 4.1 - Prob. 1ESCh. 4.1 - Prob. 2ESCh. 4.1 - Prob. 3ESCh. 4.1 - Prob. 4ESCh. 4.1 - Prob. 5ESCh. 4.1 - Prob. 6ESCh. 4.1 - Prob. 7ESCh. 4.1 - Prob. 8ESCh. 4.1 - Prob. 9ESCh. 4.1 - Prob. 10ES
Ch. 4.1 - Prob. 11ESCh. 4.1 - Prob. 12ESCh. 4.1 - Prob. 13ESCh. 4.1 - Prob. 14ESCh. 4.1 - Prob. 15ESCh. 4.1 - Prob. 16ESCh. 4.1 - Prob. 17ESCh. 4.1 - Prob. 18ESCh. 4.1 - Prob. 19ESCh. 4.1 - Prob. 20ESCh. 4.1 - Prob. 21ESCh. 4.1 - Prob. 22ESCh. 4.1 - Prob. 23ESCh. 4.1 - Prob. 24ESCh. 4.1 - Prob. 25ESCh. 4.1 - Prob. 26ESCh. 4.1 - Prob. 27ESCh. 4.1 - Prob. 28ESCh. 4.1 - Prob. 29ESCh. 4.1 - Prob. 30ESCh. 4.1 - Prob. 31ESCh. 4.1 - Prob. 32ESCh. 4.1 - Prob. 33ESCh. 4.1 - Prob. 34ESCh. 4.1 - Prob. 35ESCh. 4.1 - Prob. 36ESCh. 4.1 - Prob. 37ESCh. 4.1 - Prob. 38ESCh. 4.1 - Prob. 39ESCh. 4.1 - Prob. 40ESCh. 4.1 - Prob. 41ESCh. 4.1 - Prob. 42ESCh. 4.1 - Prob. 43ESCh. 4.1 - Prob. 44ESCh. 4.1 - Prob. 45ESCh. 4.1 - Prob. 46ESCh. 4.1 - Prob. 47ESCh. 4.1 - Prob. 48ESCh. 4.1 - Prob. 49ESCh. 4.1 - Prob. 50ESCh. 4.1 - Prob. 51ESCh. 4.1 - Prob. 52ESCh. 4.1 - Prob. 53ESCh. 4.1 - Prob. 54ESCh. 4.1 - Prob. 55ESCh. 4.1 - Prob. 56ESCh. 4.1 - Prob. 57ESCh. 4.1 - Prob. 58ESCh. 4.1 - Prob. 59ESCh. 4.1 - Prob. 60ESCh. 4.1 - Prob. 61ESCh. 4.1 - Prob. 62ESCh. 4.1 - Prob. 63ESCh. 4.2 - Prob. 1ESCh. 4.2 - Prob. 2ESCh. 4.2 - Prob. 3ESCh. 4.2 - Prob. 4ESCh. 4.2 - Prob. 5ESCh. 4.2 - Prob. 6ESCh. 4.2 - Prob. 7ESCh. 4.2 - Prob. 8ESCh. 4.2 - Prob. 9ESCh. 4.2 - Prob. 10ESCh. 4.2 - Prob. 11ESCh. 4.2 - Prob. 12ESCh. 4.2 - Prob. 13ESCh. 4.2 - Prob. 14ESCh. 4.2 - Prob. 15ESCh. 4.2 - Prob. 16ESCh. 4.2 - Prob. 17ESCh. 4.2 - Prob. 18ESCh. 4.2 - Prob. 19ESCh. 4.2 - Prob. 20ESCh. 4.2 - Prob. 21ESCh. 4.2 - Prob. 22ESCh. 4.2 - Prob. 23ESCh. 4.2 - Prob. 24ESCh. 4.2 - Prob. 25ESCh. 4.2 - Prob. 26ESCh. 4.2 - Prob. 27ESCh. 4.2 - Prob. 28ESCh. 4.2 - Prob. 29ESCh. 4.2 - Prob. 30ESCh. 4.2 - Prob. 31ESCh. 4.2 - Prob. 32ESCh. 4.2 - Prob. 33ESCh. 4.2 - Prob. 34ESCh. 4.2 - Prob. 35ESCh. 4.2 - Prob. 36ESCh. 4.2 - Prob. 37ESCh. 4.2 - Prob. 38ESCh. 4.2 - Prob. 39ESCh. 4.3 - Prob. 1ESCh. 4.3 - Prob. 2ESCh. 4.3 - Prob. 3ESCh. 4.3 - Prob. 4ESCh. 4.3 - Prob. 5ESCh. 4.3 - Prob. 6ESCh. 4.3 - Prob. 7ESCh. 4.3 - Prob. 8ESCh. 4.3 - Prob. 9ESCh. 4.3 - Prob. 10ESCh. 4.3 - Prob. 11ESCh. 4.3 - Prob. 12ESCh. 4.3 - Prob. 13ESCh. 4.3 - Prob. 14ESCh. 4.3 - Prob. 15ESCh. 4.3 - Prob. 16ESCh. 4.3 - Prob. 17ESCh. 4.3 - Prob. 18ESCh. 4.3 - Prob. 19ESCh. 4.3 - Prob. 20ESCh. 4.3 - Prob. 21ESCh. 4.3 - Prob. 22ESCh. 4.3 - Prob. 23ESCh. 4.3 - Prob. 24ESCh. 4.3 - Prob. 25ESCh. 4.3 - Prob. 26ESCh. 4.3 - Prob. 27ESCh. 4.3 - Prob. 28ESCh. 4.3 - Prob. 29ESCh. 4.3 - Prob. 30ESCh. 4.3 - Prob. 31ESCh. 4.3 - Prob. 32ESCh. 4.3 - Prob. 33ESCh. 4.3 - Prob. 34ESCh. 4.3 - Prob. 35ESCh. 4.3 - Prob. 36ESCh. 4.3 - Prob. 37ESCh. 4.3 - Prob. 38ESCh. 4.3 - Prob. 39ESCh. 4.3 - Prob. 40ESCh. 4.3 - Prob. 41ESCh. 4.3 - Prob. 42ESCh. 4.3 - Prob. 43ESCh. 4.3 - Prob. 44ESCh. 4.3 - Prob. 45ESCh. 4.3 - Prob. 46ESCh. 4.3 - Prob. 47ESCh. 4.3 - Prob. 48ESCh. 4.3 - Prob. 49ESCh. 4.4 - Prob. 1ESCh. 4.4 - Prob. 2ESCh. 4.4 - Prob. 3ESCh. 4.4 - Prob. 4ESCh. 4.4 - Prob. 5ESCh. 4.4 - Prob. 6ESCh. 4.4 - Prob. 7ESCh. 4.4 - Prob. 8ESCh. 4.4 - Prob. 9ESCh. 4.4 - Prob. 10ESCh. 4.4 - Prob. 11ESCh. 4.4 - Prob. 12ESCh. 4.4 - Prob. 13ESCh. 4.4 - Prob. 14ESCh. 4.4 - Prob. 15ESCh. 4.4 - Prob. 16ESCh. 4.4 - Prob. 17ESCh. 4.4 - Prob. 18ESCh. 4.4 - Prob. 19ESCh. 4.4 - Prob. 20ESCh. 4.4 - Prob. 21ESCh. 4.4 - Prob. 22ESCh. 4.4 - Prob. 23ESCh. 4.4 - Prob. 24ESCh. 4.4 - Prob. 25ESCh. 4.4 - Prob. 26ESCh. 4.4 - Prob. 27ESCh. 4.4 - Prob. 28ESCh. 4.4 - Prob. 29ESCh. 4.4 - Prob. 30ESCh. 4.4 - Prob. 31ESCh. 4.4 - Prob. 32ESCh. 4.4 - Prob. 33ESCh. 4.4 - Prob. 34ESCh. 4.4 - Prob. 35ESCh. 4.4 - Prob. 36ESCh. 4.4 - Prob. 37ESCh. 4.4 - Prob. 38ESCh. 4.4 - Prob. 39ESCh. 4.4 - Prob. 40ESCh. 4.4 - Prob. 41ESCh. 4.4 - Prob. 42ESCh. 4.4 - Prob. 43ESCh. 4.4 - Prob. 44ESCh. 4.4 - Prob. 45ESCh. 4.4 - Prob. 46ESCh. 4.5 - Prob. 1ESCh. 4.5 - Prob. 2ESCh. 4.5 - Prob. 3ESCh. 4.5 - Prob. 4ESCh. 4.5 - Prob. 5ESCh. 4.5 - Prob. 6ESCh. 4.5 - Prob. 7ESCh. 4.5 - Prob. 8ESCh. 4.5 - Prob. 9ESCh. 4.5 - Prob. 10ESCh. 4.5 - Prob. 11ESCh. 4.5 - Prob. 12ESCh. 4.5 - Prob. 13ESCh. 4.5 - Prob. 14ESCh. 4.5 - Prob. 15ESCh. 4.5 - Prob. 16ESCh. 4.5 - Prob. 17ESCh. 4.5 - Prob. 18ESCh. 4.5 - Prob. 19ESCh. 4.5 - Prob. 20ESCh. 4.5 - Prob. 21ESCh. 4.5 - Prob. 22ESCh. 4.5 - Prob. 23ESCh. 4.5 - Prob. 24ESCh. 4.5 - Prob. 25ESCh. 4.5 - Prob. 26ESCh. 4.5 - Prob. 27ESCh. 4.5 - Prob. 28ESCh. 4.5 - Prob. 29ESCh. 4.5 - Prob. 30ESCh. 4.5 - Prob. 31ESCh. 4.5 - Prob. 32ESCh. 4.5 - Prob. 33ESCh. 4.5 - Prob. 34ESCh. 4.5 - Prob. 35ESCh. 4.6 - Prob. 1ESCh. 4.6 - Prob. 2ESCh. 4.6 - Prob. 3ESCh. 4.6 - Prob. 4ESCh. 4.6 - Prob. 5ESCh. 4.6 - Prob. 6ESCh. 4.6 - Prob. 7ESCh. 4.6 - Prob. 8ESCh. 4.6 - Prob. 9ESCh. 4.6 - Prob. 10ESCh. 4.6 - Prob. 11ESCh. 4.6 - Prob. 12ESCh. 4.6 - Prob. 13ESCh. 4.6 - Prob. 14ESCh. 4.6 - Prob. 15ESCh. 4.6 - Prob. 16ESCh. 4.6 - Prob. 17ESCh. 4.6 - Prob. 18ESCh. 4.6 - Prob. 19ESCh. 4.6 - Prob. 20ESCh. 4.6 - Prob. 21ESCh. 4.6 - Prob. 22ESCh. 4.6 - Prob. 23ESCh. 4.6 - Prob. 24ESCh. 4.6 - Prob. 25ESCh. 4.6 - Prob. 26ESCh. 4.6 - Prob. 27ESCh. 4.6 - Prob. 28ESCh. 4.6 - Prob. 29ESCh. 4.6 - Prob. 30ESCh. 4.6 - Prob. 31ESCh. 4.6 - Prob. 32ESCh. 4.6 - Prob. 33ESCh. 4.6 - Prob. 34ESCh. 4.6 - Prob. 35ES
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- Q/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_forwardRefer 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_forward
- Refer 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_forwardRefer 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_forward
- Only 100% sure experts solve it correct complete solutions okarrow_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
- Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.arrow_forwardHomework Let X1, X2, Xn be a random sample from f(x;0) where f(x; 0) = (-), 0 < x < ∞,0 € R Using Basu's theorem, show that Y = min{X} and Z =Σ(XY) are indep. -arrow_forwardHomework Let X1, X2, Xn be a random sample from f(x; 0) where f(x; 0) = e−(2-0), 0 < x < ∞,0 € R Using Basu's theorem, show that Y = min{X} and Z =Σ(XY) are indep.arrow_forward
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