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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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 x 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
(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.)
= a. What can you conclude?

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