List all possible equivalence relations on the set (1,2,3,4) without repetition up to isomorphism. Justify your answer.
List all possible equivalence relations on the set (1,2,3,4) without repetition up to isomorphism. Justify your answer.
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
I have the following exercise:
List all possible equivalence relations on the set (1,2,3,4) without repetition up to isomorphism. Justify your answer. (Q2)
I tried to solve it and got the solution which is attached. However, I feel like something is not correct, maybe I understood the question wrong? The course is abstract algebra so help is much appreciated
![9
소
S
TA
Hence, R3 is not transitive.
✓
Ch
с
-D Since R3 is not symmetric and not transitive, R3 is not an equivalence
relation
Q2:
No. of partitions of a set: 2^-1 = 2ª - 1 =15.
Hence: {{1,2,3,43}, {{1},{2,3,4}},{{2},{1,3,4}}, {{33. {1,2,413}, {{4}, {1,2,333},
{ § 1,2}, {3,4} }, {§ 1.43,£2,33}, §§1,3}, {2,4}} §£1,2},{3}{433,
§ §1.33. {2}, {4}}.§£1.45,{2},{3}},{$13, 52167 18433.
{{1}, { 2,4}, {3}}, { {^}, {2}, {3,43}, {{1}, {2}, {3}, {4}}
3:
i) (R², $), where (x,y)* (a,b) = (x+a, y-b). (12²= {(2,W): Z, WER}
a) let (x,y),(a,b) ETR², where x, y, a, b ER
↳₂ (x+a). (y-b) → (x+a₁ y-b) € 12².
1
9
(ANSLAT
GEMEE
P
+: 0
2 معامل مد
>
219
>
+](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2521d3ac-1f2b-4213-a1c7-670c7b844752%2F62d2d458-cf06-403e-a19f-1963a0553c33%2Fld6jwfe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:9
소
S
TA
Hence, R3 is not transitive.
✓
Ch
с
-D Since R3 is not symmetric and not transitive, R3 is not an equivalence
relation
Q2:
No. of partitions of a set: 2^-1 = 2ª - 1 =15.
Hence: {{1,2,3,43}, {{1},{2,3,4}},{{2},{1,3,4}}, {{33. {1,2,413}, {{4}, {1,2,333},
{ § 1,2}, {3,4} }, {§ 1.43,£2,33}, §§1,3}, {2,4}} §£1,2},{3}{433,
§ §1.33. {2}, {4}}.§£1.45,{2},{3}},{$13, 52167 18433.
{{1}, { 2,4}, {3}}, { {^}, {2}, {3,43}, {{1}, {2}, {3}, {4}}
3:
i) (R², $), where (x,y)* (a,b) = (x+a, y-b). (12²= {(2,W): Z, WER}
a) let (x,y),(a,b) ETR², where x, y, a, b ER
↳₂ (x+a). (y-b) → (x+a₁ y-b) € 12².
1
9
(ANSLAT
GEMEE
P
+: 0
2 معامل مد
>
219
>
+
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