The 2-tuples (ordered pairs) of integers is the set Z x Z and can be given an equivalence relation under the condition (a, b)R(c, d) = ad = cb a) Explain why this forms an equivalence relation by using simple numbers in an example. You must explicitly show why this relation satisfies the properties of reflexive, symmetric, and transitive. b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x,y)], either [(a, b)] = [(x,y)] or [(a, b)] n [(x,y)] = Ø. Give at least one example for each case. Recall that [(a, b)] = {(c, d) E Z × Z: (a, b)R(c, d)}. %3D
The 2-tuples (ordered pairs) of integers is the set Z x Z and can be given an equivalence relation under the condition (a, b)R(c, d) = ad = cb a) Explain why this forms an equivalence relation by using simple numbers in an example. You must explicitly show why this relation satisfies the properties of reflexive, symmetric, and transitive. b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x,y)], either [(a, b)] = [(x,y)] or [(a, b)] n [(x,y)] = Ø. Give at least one example for each case. Recall that [(a, b)] = {(c, d) E Z × Z: (a, b)R(c, d)}. %3D
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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![The 2-tuples (ordered pairs) of integers is the set Z × Z and can be given an equivalence relation
under the condition
(a, b)R(c, d) ad =
cb
a) Explain why this forms an equivalence relation by using simple numbers in an example. You
must explicitly show why this relation satisfies the properties of reflexive, symmetric, and
transitive.
b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x, y)],
either [(a, b)] = [(x,y)] or [(a,b)] n [(x,y)] = Ø. Give at least one example for each case.
Recall that [(a, b)] = {(c, d) E Z × Z:(a, b)R(c, d)}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feefd596c-7c1c-4666-9bed-c4f5cf0c7b0d%2F81969275-1d25-4ef3-bfd2-c69349710c9b%2Feah6hrf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The 2-tuples (ordered pairs) of integers is the set Z × Z and can be given an equivalence relation
under the condition
(a, b)R(c, d) ad =
cb
a) Explain why this forms an equivalence relation by using simple numbers in an example. You
must explicitly show why this relation satisfies the properties of reflexive, symmetric, and
transitive.
b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x, y)],
either [(a, b)] = [(x,y)] or [(a,b)] n [(x,y)] = Ø. Give at least one example for each case.
Recall that [(a, b)] = {(c, d) E Z × Z:(a, b)R(c, d)}.
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