Explanation Given: { 1 , 2 , 3 , 4 } Formula Used: The given set forms group if the below four conditions are satisfied The set should be closed with respect to the operation The associative property of the operation should hold true for the elements of set The identity element should exists in the operation for all the elements of set For each element in the set has an inverse. Calculation: The set is given as { 1 , 2 , 3 , 4 } Condition 1: We have ( 1 × 1 ) mod 5 = 0 mod 5 = 0 , which belongs to { 1 , 2 , 3 , 4 } ( 1 × 2 ) mod 5 = 2 mod 5 = 2 , which belongs to { 1 , 2 , 3 , 4 } Thus, set { 1 , 2 , 3 , 4 } is closed with respect to multiplication modulo 5 Condition 2: [ [ ( 1 × 2 ) mod 5 ] × 3 ] mod 5 = [ [ 2 mod 5 ] × 3 ] mod 4 = [ 2 × 3 ] mod 5 = 6 mod 5 = 1 [ [ 1 × ( 2 × 3 ) mod 5 ] ] mod 5 = [ 1 × [ 6 mod 5 ] ] mod 5 = [ 1 × 1 ] mod 5 = 1 mod 5 = 1 Here, the associative property of multiplication modulo 5 holds true for the set { 1 , 2 , 3</

Mathematical Excursions (MindTap Course List)

4th Edition

ISBN: 9781305965584

Author: Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Clegg

Publisher: Cengage Learning

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 8.3, Problem 16ES

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Determine whether the below set forms a group with respect to the given operation.

" {1,2,3,4} ; multiplication modulo 5 "

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Chapter 8.3, Problem 15ES

Chapter 8.3, Problem 17ES

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Mathematical Excursions (MindTap Course List)

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Determine whether the below set forms a group with respect to the given operation. " { 1 , 2 , 3 , 4 } ; multiplication modulo 5 "

Solution Summary: The author explains how the set left1,2,3,4right forms a group with respect to multiplication modulo 5.

Concept explainers

Determine whether the below set forms a group with respect to the given operation.

" {1,2,3,4} ; multiplication modulo 5 "

Want to see the full answer?

Chapter 8 Solutions

Determine whether the below set forms a group with respect to the given operation. " { 1 , 2 , 3 , 4 } ; multiplication modulo 5 "

Solution Summary: The author explains how the set left1,2,3,4right forms a group with respect to multiplication modulo 5.

Concept explainers

Determine whether the below set forms a group with respect to the given operation. " {1,2,3,4} ; multiplication modulo 5 "

Want to see the full answer?

Chapter 8 Solutions

Determine whether the below set forms a group with respect to the given operation.

" {1,2,3,4} ; multiplication modulo 5 "