Chinese Remainder Theorem. Let X, Y be sets and let f: X → Y be a function. We define a relation ~ on X by declaring that a ~ b if and only if f(a) = f(b). Prove that is an equivalence relation. = Let f: X→ Y be as before, and let Q {[x] x X} denote the set of equivalence classes for the equivalence relation~ from the previous subtask. Prove that a. b. is well-defined and injective. Q→Y [a] → f(a) g:
Chinese Remainder Theorem. Let X, Y be sets and let f: X → Y be a function. We define a relation ~ on X by declaring that a ~ b if and only if f(a) = f(b). Prove that is an equivalence relation. = Let f: X→ Y be as before, and let Q {[x] x X} denote the set of equivalence classes for the equivalence relation~ from the previous subtask. Prove that a. b. is well-defined and injective. Q→Y [a] → f(a) g:
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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Proof the following step by step in detail, if able please explain why each step is taken and focus on proof writing style
![Chinese Remainder Theorem.
Let X,Y be sets and let f: X → Y be a function. We define a relation ~ on
X by declaring that a ~ b if and only if f(a) = f(b).
Prove that is an equivalence relation.
=
Let f: X→ Y be as before, and let Q {[x] xe X} denote the set of
equivalence classes for the equivalence relation ~ from the previous subtask. Prove that
a.
b.
is well-defined and injective.
g: Q→Y
[a] → f(a)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e237d3f-b8e6-4775-a6f9-5671b153aef2%2F5afe1b20-14fa-4dca-b93f-ee1dc452947e%2Fqk291h_processed.png&w=3840&q=75)
Transcribed Image Text:Chinese Remainder Theorem.
Let X,Y be sets and let f: X → Y be a function. We define a relation ~ on
X by declaring that a ~ b if and only if f(a) = f(b).
Prove that is an equivalence relation.
=
Let f: X→ Y be as before, and let Q {[x] xe X} denote the set of
equivalence classes for the equivalence relation ~ from the previous subtask. Prove that
a.
b.
is well-defined and injective.
g: Q→Y
[a] → f(a)
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