Explanation Procedure: The properties that must be satisfied by a set to be a group for modulo multiplication are: 1. Closure property: The product of any pair of elements must also be a member of the original set. 2. Associative property: For all elements, a , b and c belonging to the set, the following equation must be satisfied. ( a × b ) × c ≡ a × ( b × c ) 3. Identity property: The multiplicative identity element 1 must be in the set such that the product of any element in the set and the identity element is the same element. 4. Inverse property: For each element a that belongs to the set, there must be a unique element a’ belonging to the set such that: a a ' ≡ a ' a ≡ 1 Here, a ’ is said to be the inverse of the element a . Calculation: Consider the set { 0 , 1 , 2 , 3 } . It has to be determined whether the provided set is a group for multiplication modulo 4. So check whether all the properties are satisfied. 1. Closure property: For the provided set of elements modulo 4, the product of any pair of elements is also a member of the set. For example, 1 × 2 = 2 ≡ 2 , ( mod 4 ) 2 × 3 = 6 ≡ 2 , ( mod 4 ) So, the set is closed with respect to multiplication. 2. Associative property: The associative property is satisfied for the provided set of elements. That is, ( a × b ) × c ≡ a × ( b × c ) for all elements. So, consider three elements 1, 2, 3. from the set. So, the associative property is ( 1 × 2 ) × 3 = 1 × ( 2 × 3 )

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ISBN: 9781285697734

Author: karl J. smith

Publisher: Cengage Learning US

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Chapter 5.7, Problem 40PS

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Whether the set {0,1,2,3} is a group for multiplication modulo 4.

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Chapter 5.7, Problem 39PS

Chapter 5.7, Problem 41PS

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Whether the set { 0 , 1 , 2 , 3 } is a group for multiplication modulo 4.

Solution Summary: The author explains that the set left0,1,2,3right is not a group for multiplication modulo 4 as the inverse property isn't satisfied.

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Whether the set {0,1,2,3} is a group for multiplication modulo 4.

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Chapter 5 Solutions

Whether the set { 0 , 1 , 2 , 3 } is a group for multiplication modulo 4.

Solution Summary: The author explains that the set left0,1,2,3right is not a group for multiplication modulo 4 as the inverse property isn't satisfied.

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To find: Whether the set {0,1,2,3} is a group for multiplication modulo 4.

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Chapter 5 Solutions

To find:

Whether the set {0,1,2,3} is a group for multiplication modulo 4.