8th Edition

ISBN: 9781285463261

Author: Douglas Smith, Maurice Eggen, Richard St. Andre

Publisher: Cengage Learning

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Textbook Question

Chapter 5.1, Problem 12E

(a)Prove that ( ℤ m , + ) is associative and commutative and has an identity and that every element has an additive inverse (Theorem 6.1.2(a)).
(b)Prove that ( ℤ m , ⋅ ) is associative and commutative and has an identity when m > 1 (Theorem 6.1.2(b)).

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Chapter 5.1, Problem 11E

Chapter 5.1, Problem 13E

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Chapter 5 Solutions

A Transition to Advanced Mathematics

Ch. 5.1 - The Cayley tables for operations o,*,+, and are...Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Give an example of an algebraic structure of order...Ch. 5.1 - Prob. 6E Ch. 5.1 - Show that the structure ({1},), with operation ...Ch. 5.1 - (a)In the group G of Exercise 2, find x such that...Ch. 5.1 - Show that (,), with operation # defined by...Ch. 5.1 - Construct the operation table for each of the...

Ch. 5.1 - Prob. 11E Ch. 5.1 - (a)Prove that (m,+) is associative and commutative...Ch. 5.1 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 5.1 - Let m and a be natural numbers with am. Complete...Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Consider the set A={a,b,c,d} with operation ogiven...Ch. 5.1 - Repeat Exercise 2 with the operation * given by...Ch. 5.1 - Let m,n and M=A:A is an mn matrix with real number...Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.2 - Show that each of the following algebraic...Ch. 5.2 - Given that G={e,u,v,w} is a group of order 4 with...Ch. 5.2 - Prob. 3E Ch. 5.2 - Give an example of an algebraic system (G,o) that...Ch. 5.2 - Construct the operation table for S2. Is S2...Ch. 5.2 - Prob. 6E Ch. 5.2 - Let G be a group and aiG for all n. Prove that...Ch. 5.2 - Prove part (d) of Theorem 6.2.3. That is, prove...Ch. 5.2 - Prob. 9E Ch. 5.2 - Prob. 10E Ch. 5.2 - Prob. 11E Ch. 5.2 - Assign a grade of A (correct), C (partially...Ch. 5.3 - Assign a grade of A (correct), C (partially...Ch. 5.3 - Find all subgroups of (8,+). (U11,). (5,+). (U7,)....Ch. 5.3 - In the group S4, find two different subgroups that...Ch. 5.3 - Prove that if G is a group and H is a subgroup of...Ch. 5.3 - Prove that if H and K are subgroups of a group G,...Ch. 5.3 - Let G be a group and H be a subgroup of G. If H is...Ch. 5.3 - Prob. 7E Ch. 5.3 - Prob. 8E Ch. 5.3 - Prob. 9E Ch. 5.3 - List all generators of each cyclic group in...Ch. 5.3 - Prob. 11E Ch. 5.3 - Let G be a group, and let H be a subgroup of G....Ch. 5.3 - Let ({0},) be the group of nonzero complex numbers...Ch. 5.3 - Prob. 14E Ch. 5.3 - Prob. 15E Ch. 5.3 - Let G=a be a cyclic group of order 30. What is the...Ch. 5.4 - Is S3 isomorphic to (6,+)? Explain.Ch. 5.4 - Prob. 2E Ch. 5.4 - Use the method of proof of Cayley's Theorem to...Ch. 5.4 - Define f:++ by f(x)=x where + is the set of all...Ch. 5.4 - Assign a grade of A (correct), C (partially...Ch. 5.4 - Prob. 6E Ch. 5.4 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 5.4 - Let f the set of all real-valued integrable...Ch. 5.4 - Prob. 9E Ch. 5.4 - Find the order of each element of the group S3....Ch. 5.4 - Prob. 11E Ch. 5.4 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 5.4 - Prob. 13E Ch. 5.4 - Prob. 14E Ch. 5.4 - Prob. 15E Ch. 5.4 - Prob. 16E Ch. 5.4 - Prob. 17E Ch. 5.5 - Prob. 1E Ch. 5.5 - Prob. 2E Ch. 5.5 - Show that any two groups of order 2 are...Ch. 5.5 - Show that the function h: defined by h(x)=3x is...Ch. 5.5 - Let R be the equivalence relation on ({0}) given...Ch. 5.5 - Prob. 6E Ch. 5.5 - Prob. 7E Ch. 5.5 - Let (R,+,) be an algebraic structure such that...Ch. 5.5 - Assign a grade of A (correct), C (partially...Ch. 5.5 - Let M be the set of all 22 matrices with real...

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(a)Prove that ( ℤ m , + ) is associative and commutative and has an identity and that every element has an additive inverse (Theorem 6.1.2(a)). (b)Prove that ( ℤ m , ⋅ ) is associative and commutative and has an identity when m &gt; 1 (Theorem 6.1.2(b)).

Solution Summary: The author explains that A and B are finite disjoint sets, which are contradictory to the fact set B is infinite.

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Chapter 5 Solutions

(a)Prove that ( ℤ m , + ) is associative and commutative and has an identity and that every element has an additive inverse (Theorem 6.1.2(a)). (b)Prove that ( ℤ m , ⋅ ) is associative and commutative and has an identity when m > 1 (Theorem 6.1.2(b)).