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Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
ISBN: 9781305965584
Textbook Problem

The set of numbers { 2 k } , where k is any integer, form a group with multiplication as the operation.

a. Which element of the group is the identity element?

b. Verify that the set is closed with respect to multiplication.

c. Which element is the inverse of 2 8 ?

d. Which element is the inverse of 1 32 ?

e. In general, what is the inverse of 2 k ?

(a)

To determine

The set of numbers {2k} , where k is an integer, form a group with multiplication as the operation. Which element of the group is the identity element?

Explanation

Given: The set of numbers {2k} , where k is an integer,

Formula Used:

The given set forms group if the below four conditions are satisfied

  1. The set should be closed with respect to the operation
  2. The associative property of the operation should hold true for the elements of set
  3. The identity element should exists in the operation for all the elements of set
  4. For each element in the set has an inverse

(b)

To determine

The set of numbers {2k} , where k is an integer, form a group with multiplication as the operation. Verify that the set is closed with respect to multiplication

(c)

To determine

The set of numbers {2k} , where k is an integer, form a group with multiplication as the operation. Which element is the inverse of 28

(d)

To determine

The set of numbers {2k} , where k is an integer, form a group with multiplication as the operation. Which element is the inverse of 132 ?

(c)

To determine

The set of numbers {2k} , where k is an integer, form a group with multiplication as the operation. Which element is the inverse of 2k ?

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