Let R be a reflexive relation over a set A. Show that R is an equivalence relation if Vx,y,z E A: (x, y) E R and (x,z) ER implies that (y, z) E R. Note that we are given that Vx E A: (x, x) E R. (a) First, show that R is symmetric. Pick an ordered pair (a, b) E R. In the quantified proposition, substitute x = a, y = b,z = a. What can you conclude? (b) Next, show that R must also be transitive. (Hint: if (a, b) E R and (b, c) E R use the fact that R is symmetric and also use the quantified proposition that is given.)

Let R be a reflexive relation over a set A. Show that R is an equivalence relation if Vx,y,z E A: (x, y) E R and (x,z) ER implies that (y, z) E R. Note that we are given that Vx E A: (x, x) E R. (a) First, show that R is symmetric. Pick an ordered pair (a, b) E R. In the quantified proposition, substitute x = a, y = b,z = a. What can you conclude? (b) Next, show that R must also be transitive. (Hint: if (a, b) E R and (b, c) E R use the fact that R is symmetric and also use the quantified proposition that is given.)

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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**Title: Understanding Equivalence Relations**

Let $ R $ be a reflexive relation over a set $ A $. Show that $ R $ is an equivalence relation if $\forall x, y, z \in A: (x, y) \in R$ and $(x, z) \in R$ implies that $(y, z) \in R$.

Note that we are given that $\forall x \in A: (x, x) \in R$.

**(a)** First, show that $ R $ is symmetric. Pick an ordered pair $(a, b) \in R$. In the quantified proposition, substitute $ x = a, y = b, z = a $. What can you conclude?

**(b)** Next, show that $ R $ must also be transitive. (Hint: if $(a, b) \in R$ and $(b, c) \in R$, use the fact that $ R $ is symmetric and also use the quantified proposition that is given.)

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