   Chapter 8.3, Problem 16ES Mathematical Excursions (MindTap C...

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

Solutions

Chapter
Section 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.{ 1, 2, 3, 4); multiplication modulo 5

To determine

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

{1,2,3,4} ; multiplication modulo 5

Explanation

Given: {1,2,3,4}

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

{1,2,3,4}

Condition 1:

We have

(1×1)mod5=0mod5=0 , which belongs to {1,2,3,4}

(1×2)mod5=2mod5=2 , which belongs to {1,2,3,4}

Thus, set {1,2,3,4} is closed with respect to multiplication modulo 5

Condition 2:

[[(1×2)mod5]×3]mod5=[[2mod5]×3]mod4=[2×3]mod5=6mod5=1

[[1×(2×3)mod5]]mod5=[1×[6mod5]]mod5=[1×1]mod5=1mod5=1

Here, the associative property of multiplication modulo 5 holds true for the set {1,2,3

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