Let No = (NU{0}) be the set of natural numbers including 0. Let M = N = (No x No × No): relation - on M x M is defined as I~y (a, b, c) ~ (d, e, f) = a +b+c=d+e+f a) Prove that ~ is an equivalence relation. b) Determine all elements of the equivalence class (1,0, 1). c) Let z = (a, b, c) E M and y = (d, e, f) E M. the relation < on M x M is defined as follows: %3D I< y (a, b, c) < (d, e, f) a
Let No = (NU{0}) be the set of natural numbers including 0. Let M = N = (No x No × No): relation - on M x M is defined as I~y (a, b, c) ~ (d, e, f) = a +b+c=d+e+f a) Prove that ~ is an equivalence relation. b) Determine all elements of the equivalence class (1,0, 1). c) Let z = (a, b, c) E M and y = (d, e, f) E M. the relation < on M x M is defined as follows: %3D I< y (a, b, c) < (d, e, f) a
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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Question
![Problem 1.4: equivalence and partial order relations
Let No =
relation ~ on M x M is defined as
(NU{0}) be the set of natural numbers including 0. Let M = N = (No × No x No). he
I~y (a, b, c) ~ (d, e, f) a+b+c=d+ e+f
a) Prove that is an equivalence relation.
b) Determine all elements of the equivalence class (1,0, 1).
c) Let z (a, b, c) E M and y = (d, e, f) e M. the relation < on M xM is defined as follows:
*3 y (a, b, c) < (d, e, f) a <dbsenc<f
Prove the is a partial order.
d) Is < also a linear order? Explain why or why not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cfd2bf6-6fc3-42f6-9386-e155704e7c02%2F85b6dbe5-e6e8-4d02-a51e-a703f6b683fa%2Fnr2unuh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1.4: equivalence and partial order relations
Let No =
relation ~ on M x M is defined as
(NU{0}) be the set of natural numbers including 0. Let M = N = (No × No x No). he
I~y (a, b, c) ~ (d, e, f) a+b+c=d+ e+f
a) Prove that is an equivalence relation.
b) Determine all elements of the equivalence class (1,0, 1).
c) Let z (a, b, c) E M and y = (d, e, f) e M. the relation < on M xM is defined as follows:
*3 y (a, b, c) < (d, e, f) a <dbsenc<f
Prove the is a partial order.
d) Is < also a linear order? Explain why or why not.
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