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Mathematical Excursions (MindTap C...

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

Determine whether the set forms a group with respect to the given operation. (You may assume the operation is associative.) If the set does not form a group, determine which properties fail.

The real numbers: addition

To determine

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

“The real numbers; addition”

Explanation

Given: The set of real numbers

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.

Calculation:

The set is given as

“All real numbers”

Condition 1:

Let us suppose two real numbers as 4 and 3

Now, find 4+(3)

Since 4+(3)=1 , which is a real number

Thus, set of all real numbers is closed with respect to the operation − addition

Condition 2:

Let us consider three real numbers as 2 , 4 and 4

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