Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 1.1, Problem 10ES
Every nonzero real number has a reciropal.
- All nonzero real numbers______,
- For every nonzeros real number r, theer is ______for r,
- For every nonzero real number r, there is a real number s such that______
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7. [10 marks]
Let G
=
(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.)
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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Chapter 1 Solutions
Discrete Mathematics With Applications
Ch. 1.1 - A universal statement asserts that a certain...Ch. 1.1 - A conditional statement asserts that if one...Ch. 1.1 - Given a property that may or may not be true, an...Ch. 1.1 - In each of 1—6, fill in the blanks using a...Ch. 1.1 - In each of 1—6, fill in the blanks using a...Ch. 1.1 - In each of 1—6, fill in the blanks using a...Ch. 1.1 - Given any real number, there is a number that is...Ch. 1.1 - The reciprocal of any postive real number is...Ch. 1.1 - Prob. 6ESCh. 1.1 - Rewrite the following statements less formally,...
Ch. 1.1 - For every object J, if J is a square then J has...Ch. 1.1 - For every equation E, if E is quadratic then E has...Ch. 1.1 - Every nonzero real number has a reciropal. All...Ch. 1.1 - Evaery positive number has a positive square root....Ch. 1.1 - There is a real number whose product with every...Ch. 1.1 - There is a real number whose product with ever...Ch. 1.2 - When the elements of a set are given using the...Ch. 1.2 - The symbol R denotes ____.Ch. 1.2 - The symbol Z denotes ______Ch. 1.2 - The symbol Q denotes__Ch. 1.2 - The notation {xP(x)} is read _______Ch. 1.2 - Prob. 6TYCh. 1.2 - Prob. 7TYCh. 1.2 - Given sets A,B, and C, the Cartesian production...Ch. 1.2 - A string of length n over a set S is an ordered...Ch. 1.2 - Prob. 1ESCh. 1.2 - Write in words how to read each of the following...Ch. 1.2 - Is 4={4}? How many elements are in the set...Ch. 1.2 - a. Is 2{2}? b. How many elements are in the set...Ch. 1.2 - Which of the following sets are equal?...Ch. 1.2 - For each integer n, let Tn={n,n2} . How many...Ch. 1.2 - Prob. 7ESCh. 1.2 - Prob. 8ESCh. 1.2 - Is3{1,2,3}? Is 1{1}? Is {2}{1,2}? Is...Ch. 1.2 - Is ((2)2,22)=(22,( 2)2)? Is (5,5)=(5,5)? Is...Ch. 1.2 - Prob. 11ESCh. 1.2 - Prob. 12ESCh. 1.2 - Prob. 13ESCh. 1.2 - Prob. 14ESCh. 1.2 - Let S={0,1} . List all the string of length 4 over...Ch. 1.2 - Let T={x,y} . List all the strings of length 5...Ch. 1.3 - Given sets A and B , relation from A to B is ____Ch. 1.3 - A function F from B is a relation from A to B that...Ch. 1.3 - If F is a function from A to B and x is an element...Ch. 1.3 - Let A={2,3,4} and B={6,8,10} and define a relation...Ch. 1.3 - Let C=D={3,2,1,1,2,3} and define a elation S from...Ch. 1.3 - Let E={1,2,3} and F={2,1,0} and define a relation...Ch. 1.3 - Let G=-2,0,2) and H=4,6,8) and define a relation V...Ch. 1.3 - Define a relations S from R to R as follows: For...Ch. 1.3 - Define a relation R from R to R as follows: For...Ch. 1.3 - Let A={4,5,6} and B={5,6,7} and define relations...Ch. 1.3 - Let A={2,4} and B={1,3,5} and define relations U,...Ch. 1.3 - Find all function from {01,} to {1} . Find two...Ch. 1.3 - Find tour relations from {a,b} to {x,y} that are...Ch. 1.3 - Let A={0,1,2} and let S be the set of all strings...Ch. 1.3 - Let A={x,y} and let S be the set all strings over...Ch. 1.3 - Let A={1,0,1} and B={t,u,v,w} . Define a function...Ch. 1.3 - Let C = (1,2,3,4) and D={a,b,c,d}. Define a...Ch. 1.3 - Let X=2,4,5) and Y=(1,2,4,6) . Which of the...Ch. 1.3 - Let f be the squaring function defined in Example...Ch. 1.3 - Let g be the successor function defined in Example...Ch. 1.3 - Let h be the constant function defined in Example...Ch. 1.3 - Define functions f and g from R to R by the...Ch. 1.3 - Define functions H and K from R to R by the...Ch. 1.4 - A graph consists of two finite sets: ______and...Ch. 1.4 - A loop in a graph is_____Ch. 1.4 - Two distinct edges in a graph are parallel if, and...Ch. 1.4 - Two vertices are called adjacent if, and only if,...Ch. 1.4 - An edge is incident on _______Ch. 1.4 - Two edges incident on the same endpoint...Ch. 1.4 - A vertex on which no edges are incident is________Ch. 1.4 - Prob. 8TYCh. 1.4 - Prob. 9TYCh. 1.4 - In 1 and 2, graphs are represented by drawings...Ch. 1.4 - In 1 and 2, graphs are represented by drawings....Ch. 1.4 - In 3 and 4, draw pictures of the specified graphs....Ch. 1.4 - Prob. 4ESCh. 1.4 - Prob. 5ESCh. 1.4 - In 5-7, show that the two drawings represent the...Ch. 1.4 - In 5-7, show that the two drawings represent the...Ch. 1.4 - For each of the graphs in 8 and 9: (i) Find all...Ch. 1.4 - For each of the graphs in 8 and 9: (i) Find all...Ch. 1.4 - Use the graph of Example 1.4.6 to determine...Ch. 1.4 - Find three other winning sequences of moves for...Ch. 1.4 - Another famous puzzle used as an example in the...Ch. 1.4 - Solve the vegetarians-and-cannibals puzzle for the...Ch. 1.4 - Two jugs A and B have capacities of 3 quarts and 5...Ch. 1.4 - Prob. 15ESCh. 1.4 - In this exercise a graph is used to help solve a...Ch. 1.4 - A deptnn1 war to ithechik final ezans that no...
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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_forwardOnly 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_forward
- 3. [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_forwardLet 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_forward
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