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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 rational numbers: addition (Recall that a rational number is any number that can be expressed in the form p/q, where p and q are integers. q

0.)

To determine

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

“The rational numbers; addition”

Explanation

Given: The set of rational 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 rational numbers”

Condition 1:

Let us suppose two rational as 4 and 3

Now, find 4+(3)

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

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

Condition 2:

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

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