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Math Discrete Mathematics With Applications

As in Example 8.1.2, the congruence modulo 2 relation E is defined from Z to Z as follows: For every ordered ( m , n ) ∈ Z × Z . m E n ⇔ m − n is even . Is 0 E 0? Is 5 E 2? Is (6,6) ∈ E ? Prove that for any even integer n , n E 0 .

As in Example 8.1.2, the congruence modulo 2 relation E is defined from Z to Z as follows: For every ordered ( m , n ) ∈ Z × Z . m E n ⇔ m − n is even . Is 0 E 0? Is 5 E 2? Is (6,6) ∈ E ? Prove that for any even integer n , n E 0 .

Solution Summary: The author calculates whether 0 E 0, 5 E 2, 6, 6) E and (-1, 7) - E are true or not for the relation E.

Discrete Mathematics With Applications

Discrete Mathematics With Applications

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Textbook Question

Chapter 8.1, Problem 1ES

As in Example 8.1.2, the congruence modulo 2 relation E is defined from Z to Z as follows: For every ordered ( m , n ) ∈ Z × Z .

m E n ⇔ m − n is even .

Is 0 E 0? Is 5 E 2? Is (6,6) ∈ E ?
Prove that for any even integer n , n E 0 .

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Chapter 8.1, Problem 5TY

Chapter 8.1, Problem 2ES

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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. 2TY Ch. 8.1 - Prob. 3TY Ch. 8.1 - Prob. 4TY Ch. 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. 8ES Ch. 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. 12ES Ch. 8.1 - Prob. 13ES Ch. 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. 16ES Ch. 8.1 - Prob. 17ES Ch. 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. 20ES Ch. 8.1 - Define relation R and S on R as follows:...Ch. 8.1 - Prob. 22ES Ch. 8.1 - Prob. 23ES Ch. 8.1 - Prob. 24ES Ch. 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. 4TY Ch. 8.2 - Prob. 5TY Ch. 8.2 - Prob. 6TY Ch. 8.2 - Prob. 7TY Ch. 8.2 - Prob. 8TY Ch. 8.2 - Prob. 9TY Ch. 8.2 - Prob. 10TY Ch. 8.2 - Prob. 1ES Ch. 8.2 - In 1-8, a number of relations are defined on the...Ch. 8.2 - Prob. 3ES Ch. 8.2 - Prob. 4ES 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 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. 15ES Ch. 8.2 - Prob. 16ES Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 18ES Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 20ES Ch. 8.2 - Prob. 21ES 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. 24ES Ch. 8.2 - In 9-33, determine whether the given is reflexive...Ch. 8.2 - Prob. 26ES Ch. 8.2 - Prob. 27ES Ch. 8.2 - Prob. 28ES Ch. 8.2 - Prob. 29ES Ch. 8.2 - In 9-33, determine whether the given relation is...Ch. 8.2 - Prob. 31ES 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 34-36, assume that R is a relation on a et A....Ch. 8.2 - Prob. 35ES Ch. 8.2 - Prob. 36ES Ch. 8.2 - Prob. 37ES Ch. 8.2 - Prob. 38ES Ch. 8.2 - Prob. 39ES Ch. 8.2 - Prob. 40ES Ch. 8.2 - Prob. 41ES Ch. 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. 44ES Ch. 8.2 - Prob. 45ES Ch. 8.2 - Prob. 46ES Ch. 8.2 - Prob. 47ES Ch. 8.2 - In 43-50, the following definitions are used: A...Ch. 8.2 - Prob. 49ES Ch. 8.2 - Prob. 50ES Ch. 8.2 - Prob. 51ES Ch. 8.2 - In 51—53, R, S, and T are relations defined on...Ch. 8.2 - Prob. 53ES Ch. 8.2 - Prob. 54ES Ch. 8.2 - Prob. 55ES Ch. 8.2 - Prob. 56ES Ch. 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. 3TY Ch. 8.3 - Prob. 4TY Ch. 8.3 - Prob. 5TY Ch. 8.3 - Prob. 6TY Ch. 8.3 - Prob. 1ES Ch. 8.3 - Prob. 2ES Ch. 8.3 - Prob. 3ES Ch. 8.3 - In each of 3—6, the relation R is an equivalence...Ch. 8.3 - Prob. 5ES Ch. 8.3 - In each of 3-6, the relation R is an equivalence...Ch. 8.3 - Prob. 7ES Ch. 8.3 - Prob. 8ES Ch. 8.3 - Prob. 9ES Ch. 8.3 - In each of 7-14, relation R is an equivalence...Ch. 8.3 - Prob. 11ES Ch. 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. 17ES Ch. 8.3 - Prob. 18ES Ch. 8.3 - In 19-31, (1) prove that the relation is an...Ch. 8.3 - Prob. 20ES Ch. 8.3 - Prob. 21ES Ch. 8.3 - Prob. 22ES Ch. 8.3 - Prob. 23ES Ch. 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. 26ES Ch. 8.3 - Prob. 27ES Ch. 8.3 - Prob. 28ES Ch. 8.3 - Prob. 29ES Ch. 8.3 - Prob. 30ES Ch. 8.3 - In 19—31, (1) prove that the relation is an...Ch. 8.3 - Prob. 32ES Ch. 8.3 - Prob. 33ES Ch. 8.3 - Prob. 34ES Ch. 8.3 - Prob. 35ES Ch. 8.3 - Prob. 36ES Ch. 8.3 - Prob. 37ES Ch. 8.3 - Prob. 38ES Ch. 8.3 - Prob. 39ES Ch. 8.3 - Prob. 40ES Ch. 8.3 - Prob. 41ES Ch. 8.3 - Prob. 42ES Ch. 8.3 - Prob. 43ES Ch. 8.3 - Let A=Z+Z+ . Define a relation R on A as follows:...Ch. 8.3 - Prob. 45ES Ch. 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. 2TY Ch. 8.4 - Prob. 3TY Ch. 8.4 - Prob. 4TY Ch. 8.4 - Prob. 5TY Ch. 8.4 - Prob. 6TY Ch. 8.4 - Prob. 7TY Ch. 8.4 - Prob. 8TY Ch. 8.4 - Fermat’s little theorem says that if p is any...Ch. 8.4 - Prob. 10TY Ch. 8.4 - Prob. 1ES Ch. 8.4 - Use the Caesar cipher to encrypt the message AN...Ch. 8.4 - Prob. 3ES Ch. 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. 6ES Ch. 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. 9ES Ch. 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. 13ES Ch. 8.4 - Prob. 14ES Ch. 8.4 - Prob. 15ES Ch. 8.4 - In 16-18, use the techniques of Example 8.4.4 and...Ch. 8.4 - Prob. 17ES Ch. 8.4 - Prob. 18ES Ch. 8.4 - Prob. 19ES Ch. 8.4 - Prob. 20ES Ch. 8.4 - Prob. 21ES Ch. 8.4 - In 19-24, use the RSA cipher from Examples 8.4.9...Ch. 8.4 - Prob. 23ES Ch. 8.4 - Prob. 24ES Ch. 8.4 - Prob. 25ES Ch. 8.4 - Prob. 26ES Ch. 8.4 - In 26 and 27, use the extended Euclidean algorithm...Ch. 8.4 - Prob. 28ES Ch. 8.4 - Prob. 29ES Ch. 8.4 - Prob. 30ES Ch. 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. 33ES Ch. 8.4 - Prob. 34ES Ch. 8.4 - Prob. 35ES Ch. 8.4 - In 36,37,39 and 40, use the RSA cipher with public...Ch. 8.4 - Prob. 37ES Ch. 8.4 - Find the least positive inverse for 43 modulo 660.Ch. 8.4 - Prob. 39ES Ch. 8.4 - Prob. 40ES Ch. 8.4 - Prob. 41ES Ch. 8.4 - Prob. 42ES Ch. 8.4 - Prob. 43ES Ch. 8.5 - Prob. 1TY Ch. 8.5 - Prob. 2TY Ch. 8.5 - Prob. 3TY Ch. 8.5 - Prob. 4TY Ch. 8.5 - Prob. 5TY Ch. 8.5 - Prob. 6TY Ch. 8.5 - Prob. 7TY Ch. 8.5 - Prob. 8TY Ch. 8.5 - Prob. 9TY Ch. 8.5 - Prob. 10TY Ch. 8.5 - Each of the following is a relation on {0,1,2,3}...Ch. 8.5 - Prob. 2ES Ch. 8.5 - Let S be the set of all strings of a’s and b’s....Ch. 8.5 - Prob. 4ES Ch. 8.5 - Prob. 5ES Ch. 8.5 - Let P be the set of all people who have ever lived...Ch. 8.5 - Prob. 7ES Ch. 8.5 - Prob. 8ES Ch. 8.5 - Prob. 9ES Ch. 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. 12ES Ch. 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. 15ES Ch. 8.5 - Consider the “divides” relation on each of the...Ch. 8.5 - Prob. 17ES 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 - Let S={0,1} and consider the partial order...Ch. 8.5 - Consider the “divides” relation defined on the set...Ch. 8.5 - Prob. 22ES Ch. 8.5 - Prob. 23ES Ch. 8.5 - Prob. 24ES Ch. 8.5 - Prob. 25ES Ch. 8.5 - Prob. 26ES Ch. 8.5 - Prob. 27ES Ch. 8.5 - Prob. 28ES Ch. 8.5 - Prob. 29ES Ch. 8.5 - Prob. 30ES Ch. 8.5 - Prob. 31ES Ch. 8.5 - Prob. 32ES Ch. 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. 35ES Ch. 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. 38ES Ch. 8.5 - Prob. 39ES Ch. 8.5 - Prob. 40ES Ch. 8.5 - Prob. 41ES Ch. 8.5 - Prob. 42ES Ch. 8.5 - Prob. 43ES Ch. 8.5 - Prob. 44ES Ch. 8.5 - Prob. 45ES Ch. 8.5 - Prob. 46ES Ch. 8.5 - Prob. 47ES Ch. 8.5 - Prob. 48ES Ch. 8.5 - Prob. 49ES Ch. 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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