**Transcription for Educational Website:**6. Let $ R $ be the partial order which has the Hasse diagram shown. List the elements of $ R $.**Description of the Hasse Diagram:**- The diagram contains three elements: $ a $, $ b $, and $ c $.- Element $ a $ is at the bottom of the diagram.- Both $ b $ and $ c $ are directly above $ a $, indicating that $ a $ is related by the partial order to both $ b $ and $ c $ (i.e., $ a \leq b $ and $ a \leq c $).- There is an additional element $ d $ positioned above both $ b $ and $ c $, indicating that $ b \leq d $ and $ c \leq d $.**Elements of $ R $:**- The set $ R $ includes the elements $ a $, $ b $, $ c $, and $ d $ with the relations described by the Hasse diagram.

R is a partial order relation means, R is reflexive, anti-symmetric & transitive. If R is a partial order on a finite set A, we construct a Hasse diagram for R on A by drawing a line from x up to y, if x,y∈A with xRy & most important, if there is no other element z∈A such that xRz and zRy.(so there is nothing in between x and y).If we adopt the convention of reading the diagram from bottom to top, then it is not necessary to direct any edge. Also there ids no edge showing the relation xRx. Here given the Hasse diagram of the relation R . To find the relation R: Read the Hasse diagram from bottom to top we obtain the , aRb⇒a,b∈RbRd ⇒b,d∈RaRc ⇒a,c∈R Also R is reflexive( Since it is a Partial order), thus there must exist a,a,b,b,c,c,d,d Also R is transitive( Since R is a Partial order), thus aRb & bRd⇒aRd⇒a,d∈R. Thus, relation R is, R=a,a,a,b,a,c,a,d,b,b,b,d,c,c,d,d

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6. Let R be the partial order which has the Hasse diagram shown. List the elements of R.

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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**Transcription for Educational Website:**

6. Let $ R $ be the partial order which has the Hasse diagram shown. List the elements of $ R $.

**Description of the Hasse Diagram:**

- The diagram contains three elements: $ a $, $ b $, and $ c $.
- Element $ a $ is at the bottom of the diagram.
- Both $ b $ and $ c $ are directly above $ a $, indicating that $ a $ is related by the partial order to both $ b $ and $ c $ (i.e., $ a \leq b $ and $ a \leq c $).
- There is an additional element $ d $ positioned above both $ b $ and $ c $, indicating that $ b \leq d $ and $ c \leq d $.

**Elements of $ R $:**

- The set $ R $ includes the elements $ a $, $ b $, $ c $, and $ d $ with the relations described by the Hasse diagram.

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