Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in correct order to show that R is an equivalence relation. (Prove the given relation is reflexive first, and then symmetric and transitive.) Rank the options below. Hence, R is reflexive. If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de. Hence, Ris transitive. 60 7 3 Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R. 2 If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R. Hence, Ris symmetric. ((a, b), (a, b)) E E R because ab = ba. Since R is reflexive, symmetric, and transitive, it is an equivalence relation. 8 00 5 4
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in correct order to show that R is an equivalence relation. (Prove the given relation is reflexive first, and then symmetric and transitive.) Rank the options below. Hence, R is reflexive. If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de. Hence, Ris transitive. 60 7 3 Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R. 2 If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R. Hence, Ris symmetric. ((a, b), (a, b)) E E R because ab = ba. Since R is reflexive, symmetric, and transitive, it is an equivalence relation. 8 00 5 4
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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![Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in
correct order to show that R is an equivalence relation.
(Prove the given relation is reflexive first, and then symmetric and transitive.)
Rank the options below.
Hence, R is reflexive.
If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de.
Hence, Ris transitive.
60
7
3
Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R.
2
If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R.
Hence, Ris symmetric.
((a, b), (a, b)) E E R because ab = ba.
Since R is reflexive, symmetric, and transitive, it is an equivalence relation.
8
00
5
4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8be5dca0-d576-47ed-9914-a2624fdc4b91%2Fab1852e6-4922-4049-ad4a-ad36dd7295e8%2Fs8win93_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in
correct order to show that R is an equivalence relation.
(Prove the given relation is reflexive first, and then symmetric and transitive.)
Rank the options below.
Hence, R is reflexive.
If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de.
Hence, Ris transitive.
60
7
3
Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R.
2
If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R.
Hence, Ris symmetric.
((a, b), (a, b)) E E R because ab = ba.
Since R is reflexive, symmetric, and transitive, it is an equivalence relation.
8
00
5
4
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