Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 8.3, Problem 3TY
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Refer to page 12 for a problem on solving a homogeneous differential equation.
Instructions:
• Simplify the equation into a homogeneous form.
Use appropriate substitutions to reduce complexity.
Solve systematically and verify the final result with clear back-substitutions.
Link:
[https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]
Refer to page 36 for solving a bang-bang control problem.
Instructions:
•
Formulate the problem, identifying the control constraints.
•
•
Apply Pontryagin's Maximum Principle to derive the switching conditions.
Clearly illustrate the switching points in the control trajectory. Verify the solution satisfies the
optimality criteria.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]
Total marks 16
5.
Let (N,F,P) be a probability space and let X : N → R be a
random variable such that the probability density function is given by
f(x)=ex for x € R.
(i)
Find the characteristic function of the random variable X.
[8 Marks]
(ii) Using the result of (i), calculate the first two moments of
the random variable X, i.e., E(X") for n = 1,2.
(iii)
What is the variance of X.
[6 Marks]
[2 Marks]
Chapter 8 Solutions
Discrete Mathematics With Applications
Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TYCh. 8.1 - Prob. 3TYCh. 8.1 - Prob. 4TYCh. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES
Ch. 8.1 - Let X={a,b,c}. Define a relation J on P(X) as...Ch. 8.1 - Define a relation R on Z as follows: For all...Ch. 8.1 - Prob. 8ESCh. 8.1 - Let A be the set of all strings of 0’s, 1’s, and...Ch. 8.1 - Let A={3,4,5} and B={4,5,6} and let R be the “less...Ch. 8.1 - Let A={3,4,5} and B={4,5,6} and let S be the...Ch. 8.1 - Prob. 12ESCh. 8.1 - Prob. 13ESCh. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Prob. 16ESCh. 8.1 - Prob. 17ESCh. 8.1 - Draw the directed graphs of the relations defined...Ch. 8.1 - Exercises 19-20 refer to unions and intersections...Ch. 8.1 - Prob. 20ESCh. 8.1 - Define relation R and S on R as follows:...Ch. 8.1 - Prob. 22ESCh. 8.1 - Prob. 23ESCh. 8.1 - Prob. 24ESCh. 8.2 - For a relation R on a set A to be reflexive means...Ch. 8.2 - For a relation R on a set A to be symmetric means...Ch. 8.2 - For a relation R on a set A to be transitive means...Ch. 8.2 - Prob. 4TYCh. 8.2 - Prob. 5TYCh. 8.2 - Prob. 6TYCh. 8.2 - Prob. 7TYCh. 8.2 - Prob. 8TYCh. 8.2 - Prob. 9TYCh. 8.2 - Prob. 10TYCh. 8.2 - Prob. 1ESCh. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - Prob. 3ESCh. 8.2 - Prob. 4ESCh. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9—33, determine whether the given relation is...Ch. 8.2 - In 9—33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 15ESCh. 8.2 - Prob. 16ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 18ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 20ESCh. 8.2 - Prob. 21ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 24ESCh. 8.2 - In 9-33, determine whether the given is reflexive...Ch. 8.2 - Prob. 26ESCh. 8.2 - Prob. 27ESCh. 8.2 - Prob. 28ESCh. 8.2 - Prob. 29ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 31ESCh. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - In 34-36, assume that R is a relation on a et A....Ch. 8.2 - Prob. 35ESCh. 8.2 - Prob. 36ESCh. 8.2 - Prob. 37ESCh. 8.2 - Prob. 38ESCh. 8.2 - Prob. 39ESCh. 8.2 - Prob. 40ESCh. 8.2 - Prob. 41ESCh. 8.2 - In 37-42, assume that R and S are relations on a...Ch. 8.2 - In 43-50, the following definitions are used: A...Ch. 8.2 - Prob. 44ESCh. 8.2 - Prob. 45ESCh. 8.2 - Prob. 46ESCh. 8.2 - Prob. 47ESCh. 8.2 - In 43-50, the following definitions are used: A...Ch. 8.2 - Prob. 49ESCh. 8.2 - Prob. 50ESCh. 8.2 - Prob. 51ESCh. 8.2 - In 51—53, R, S, and T are relations defined on...Ch. 8.2 - Prob. 53ESCh. 8.2 - Prob. 54ESCh. 8.2 - Prob. 55ESCh. 8.2 - Prob. 56ESCh. 8.3 - For a relation on a set to be an equivalence...Ch. 8.3 - The notation m=n(modd) is...Ch. 8.3 - Prob. 3TYCh. 8.3 - Prob. 4TYCh. 8.3 - Prob. 5TYCh. 8.3 - Prob. 6TYCh. 8.3 - Prob. 1ESCh. 8.3 - Prob. 2ESCh. 8.3 - Prob. 3ESCh. 8.3 - In each of 3—6, the relation R is an equivalence...Ch. 8.3 - Prob. 5ESCh. 8.3 - In each of 3-6, the relation R is an equivalence...Ch. 8.3 - Prob. 7ESCh. 8.3 - Prob. 8ESCh. 8.3 - Prob. 9ESCh. 8.3 - In each of 7-14, relation R is an equivalence...Ch. 8.3 - Prob. 11ESCh. 8.3 - In each of 7-14, relation R is an equivalence...Ch. 8.3 - In each of 7-14, the relation R is an equivalence...Ch. 8.3 - In each of 7—14, the relation R is an equivalence...Ch. 8.3 - Determine which of the following congruence...Ch. 8.3 - Let R be the relation of congruence modulo 3....Ch. 8.3 - Prob. 17ESCh. 8.3 - Prob. 18ESCh. 8.3 - In 19-31, (1) prove that the relation is an...Ch. 8.3 - Prob. 20ESCh. 8.3 - Prob. 21ESCh. 8.3 - Prob. 22ESCh. 8.3 - Prob. 23ESCh. 8.3 - In 19-31. (1) prove that the relation is an...Ch. 8.3 - In 19-31,(1) prove that the relation is an...Ch. 8.3 - Prob. 26ESCh. 8.3 - Prob. 27ESCh. 8.3 - Prob. 28ESCh. 8.3 - Prob. 29ESCh. 8.3 - Prob. 30ESCh. 8.3 - In 19—31, (1) prove that the relation is an...Ch. 8.3 - Prob. 32ESCh. 8.3 - Prob. 33ESCh. 8.3 - Prob. 34ESCh. 8.3 - Prob. 35ESCh. 8.3 - Prob. 36ESCh. 8.3 - Prob. 37ESCh. 8.3 - Prob. 38ESCh. 8.3 - Prob. 39ESCh. 8.3 - Prob. 40ESCh. 8.3 - Prob. 41ESCh. 8.3 - Prob. 42ESCh. 8.3 - Prob. 43ESCh. 8.3 - Let A=Z+Z+ . Define a relation R on A as follows:...Ch. 8.3 - Prob. 45ESCh. 8.3 - Let R be a relation on a set A and suppose R is...Ch. 8.3 - Refer to the quote at the beginning of this...Ch. 8.4 - When letters of the alphabet are encrypted using...Ch. 8.4 - Prob. 2TYCh. 8.4 - Prob. 3TYCh. 8.4 - Prob. 4TYCh. 8.4 - Prob. 5TYCh. 8.4 - Prob. 6TYCh. 8.4 - Prob. 7TYCh. 8.4 - Prob. 8TYCh. 8.4 - Fermat’s little theorem says that if p is any...Ch. 8.4 - Prob. 10TYCh. 8.4 - Prob. 1ESCh. 8.4 - Use the Caesar cipher to encrypt the message AN...Ch. 8.4 - Prob. 3ESCh. 8.4 - Let a=68, b=33, and n=7. Verify that 7|(68-33)....Ch. 8.4 - Prove the transitivity of modular congruence. That...Ch. 8.4 - Prob. 6ESCh. 8.4 - Verify the following statements. 128=2(mod7) and...Ch. 8.4 - Verify the following statements. 45=3 (mod 6) and...Ch. 8.4 - Prob. 9ESCh. 8.4 - In 9-11, prove each of the given statements,...Ch. 8.4 - In 9-11, prove each of the given statements,...Ch. 8.4 - Prove that for every integer n0,10n=1(mod9) . Use...Ch. 8.4 - Prob. 13ESCh. 8.4 - Prob. 14ESCh. 8.4 - Prob. 15ESCh. 8.4 - In 16-18, use the techniques of Example 8.4.4 and...Ch. 8.4 - Prob. 17ESCh. 8.4 - Prob. 18ESCh. 8.4 - Prob. 19ESCh. 8.4 - Prob. 20ESCh. 8.4 - Prob. 21ESCh. 8.4 - In 19-24, use the RSA cipher from Examples 8.4.9...Ch. 8.4 - Prob. 23ESCh. 8.4 - Prob. 24ESCh. 8.4 - Prob. 25ESCh. 8.4 - Prob. 26ESCh. 8.4 - In 26 and 27, use the extended Euclidean algorithm...Ch. 8.4 - Prob. 28ESCh. 8.4 - Prob. 29ESCh. 8.4 - Prob. 30ESCh. 8.4 - Find an inverse for 210 modulo 13. Find appositive...Ch. 8.4 - Find an inverse for 41 modulo 660. Find the least...Ch. 8.4 - Prob. 33ESCh. 8.4 - Prob. 34ESCh. 8.4 - Prob. 35ESCh. 8.4 - In 36,37,39 and 40, use the RSA cipher with public...Ch. 8.4 - Prob. 37ESCh. 8.4 - Find the least positive inverse for 43 modulo 660.Ch. 8.4 - Prob. 39ESCh. 8.4 - Prob. 40ESCh. 8.4 - Prob. 41ESCh. 8.4 - Prob. 42ESCh. 8.4 - Prob. 43ESCh. 8.5 - Prob. 1TYCh. 8.5 - Prob. 2TYCh. 8.5 - Prob. 3TYCh. 8.5 - Prob. 4TYCh. 8.5 - Prob. 5TYCh. 8.5 - Prob. 6TYCh. 8.5 - Prob. 7TYCh. 8.5 - Prob. 8TYCh. 8.5 - Prob. 9TYCh. 8.5 - Prob. 10TYCh. 8.5 - Each of the following is a relation on {0,1,2,3}...Ch. 8.5 - Prob. 2ESCh. 8.5 - Let S be the set of all strings of a’s and b’s....Ch. 8.5 - Prob. 4ESCh. 8.5 - Prob. 5ESCh. 8.5 - Let P be the set of all people who have ever lived...Ch. 8.5 - Prob. 7ESCh. 8.5 - Prob. 8ESCh. 8.5 - Prob. 9ESCh. 8.5 - Suppose R and S are antisymmetric relations on a...Ch. 8.5 - Let A={a,b}, and supposeAhas the partial order...Ch. 8.5 - Prob. 12ESCh. 8.5 - Let A={a,b} . Describe all partial order relations...Ch. 8.5 - Let A={a,b,c}. Describe all partial order...Ch. 8.5 - Prob. 15ESCh. 8.5 - Consider the “divides” relation on each of the...Ch. 8.5 - Prob. 17ESCh. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Let S={0,1} and consider the partial order...Ch. 8.5 - Consider the “divides” relation defined on the set...Ch. 8.5 - Prob. 22ESCh. 8.5 - Prob. 23ESCh. 8.5 - Prob. 24ESCh. 8.5 - Prob. 25ESCh. 8.5 - Prob. 26ESCh. 8.5 - Prob. 27ESCh. 8.5 - Prob. 28ESCh. 8.5 - Prob. 29ESCh. 8.5 - Prob. 30ESCh. 8.5 - Prob. 31ESCh. 8.5 - Prob. 32ESCh. 8.5 - Consider the set A={12,24,48,3,9} ordered by the...Ch. 8.5 - Suppose that R is a partial order relation on a...Ch. 8.5 - Prob. 35ESCh. 8.5 - The set A={2,4,3,6,12,18,24} is partially ordered...Ch. 8.5 - Find a chain of length 2 for the relation defined...Ch. 8.5 - Prob. 38ESCh. 8.5 - Prob. 39ESCh. 8.5 - Prob. 40ESCh. 8.5 - Prob. 41ESCh. 8.5 - Prob. 42ESCh. 8.5 - Prob. 43ESCh. 8.5 - Prob. 44ESCh. 8.5 - Prob. 45ESCh. 8.5 - Prob. 46ESCh. 8.5 - Prob. 47ESCh. 8.5 - Prob. 48ESCh. 8.5 - Prob. 49ESCh. 8.5 - A set S of jobs can be ordered by writing x_y to...Ch. 8.5 - Suppose the tasks described in Example 8.5.12...
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- 6. Let X be a random variable taking values in (0,∞) with proba- bility density function fx(u) = 5e5u u > 0. Total marks 8 Let Y = X2. Find the probability density function of Y. [8 Marks]arrow_forward5. Let a probability measure P on ([0,3], B([0,3])) be given by 1 dP(s): = ½ s² ds. 9 Consider a random variable X : [0,3] → R given by X(s) = s², sc [0,3]. S Total marks 7 Find the distribution of X. [7 Marks]arrow_forwardRefer to page 24 for solving a differential equation using Laplace transforms. Instructions: Take the Laplace transform of the given equation, applying initial conditions appropriately. ⚫ Solve the resulting algebraic equation and find the inverse transform. Provide step-by-step solutions with intermediate results and final verification. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
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