### Relations and Equivalence Relations#### Question:**2. Explain why the following binary relation on S = {1, 2, 3} is not an equivalence relation on S.**The relation $ R $ is defined as follows: \[ R = \{ (1,1), (1,2), (1,3), (2,1), (2,2), (3,3) \} \]#### Solution:To determine why $ R $ is not an equivalence relation on the set $ S = \{1, 2, 3\} $, we must evaluate it with respect to the three properties of equivalence relations:1. **Reflexivity**: - For a relation to be reflexive, every element must be related to itself. Specifically, for all $ x \in S $, the pair $ (x, x) $ must be in $ R $. - In this case, the pairs $ (1,1) $, $ (2,2) $, and $ (3,3) $ are present in $ R $, so the relation is reflexive.2. **Symmetry**: - For a relation to be symmetric, if $ (a,b) \in R $, then $ (b,a) $ must also be in $ R $. - Here, we see that both $ (1,2) $ and $ (2,1) $ are present, which satisfies the symmetric property for these two pairs. However, $ (1,3) $ is in $ R $ but $ (3,1) $ is not in $ R $. This violates the symmetry requirement.3. **Transitivity**: - For a relation to be transitive, if $ (a,b) \in R $ and $ (b,c) \in R $, then $ (a,c) $ must also be in $ R $. - In this set, for example, $ (1, 2) \in R $ and $ (2,1) \in R $ imply that $ (1,1) \in R $, which is true. However, $ (1,2) \in R $ and $ (2,3) \in R $ should imply \( (1,3)

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Explain whny the following binary relation on. 1,2,3} Is not R= {(1,1)(1,2),(1,3),(2,1).(2,2).(3,3)}|

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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