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Determine whether the following mathematical systems are groups or not. If it's a group, explain why all four group conditions (closure, identity, inverses, associativity) are satisfied. Also, specify if the group is Abelian (commutative). If it's not a group, show at least one of the four group conditions that is not satisfied. a) The set of even whole numbers under addition. (Remember that the number 0 is even since it is a multiple of 2:0 = 2 x 0)

Determine whether the following mathematical systems are groups or not. If it's a group, explain why all four group conditions (closure, identity, inverses, associativity) are satisfied. Also, specify if the group is Abelian (commutative). If it's not a group, show at least one of the four group conditions that is not satisfied. a) The set of even whole numbers under addition. (Remember that the number 0 is even since it is a multiple of 2:0 = 2 x 0)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 42E: 42. For an arbitrary set , the power set was defined in Section by , and addition in was...
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Problem 1 Determine whether the following mathematical systems are groups or not. If it's a group, explain why all four group conditions (closure, identity, inverses, associativity) are satisfied. Also, specify if the group is Abelian (commutative). If it's not a group, show at least one of the four group conditions that is not satisfied. a) The set of even whole numbers under addition. (Remember that the number 0 is even since it is a multiple of 2:0 = 2 ×0)
Transcribed Image Text:Problem 1 Determine whether the following mathematical systems are groups or not. If it's a group, explain why all four group conditions (closure, identity, inverses, associativity) are satisfied. Also, specify if the group is Abelian (commutative). If it's not a group, show at least one of the four group conditions that is not satisfied. a) The set of even whole numbers under addition. (Remember that the number 0 is even since it is a multiple of 2:0 = 2 ×0)
b) The set of positive fractions under multiplication. c) The set of integers under the averaging operation, which takes the sum of any two integers in this set and divides it by 2.
Transcribed Image Text:b) The set of positive fractions under multiplication. c) The set of integers under the averaging operation, which takes the sum of any two integers in this set and divides it by 2.
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