(a) F(n) = n³ (b) F(n) = n mod 2 (c) F(n) = n mod 3 (d) F(n) = the largest prime factor n has. (e) F(n) = 331
(a) F(n) = n³ (b) F(n) = n mod 2 (c) F(n) = n mod 3 (d) F(n) = the largest prime factor n has. (e) F(n) = 331
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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Abstract Algebra
![**Warmup for Next Week**
For each integer \( n \), define \([n] = n + 6\mathbb{Z}\), which represents the left coset of \( \mathbb{Z} \) by \( n \). Let \( \Omega = \{[0], [1], [2], [3], [4], [5]\} \) be the set of cosets.
A function \( F \), with domain \( \mathbb{Z} \), is considered **constant on cosets** if, for all \( n_1, n_2 \), if \([n_1] = [n_2]\), then \( F([n_1]) = F([n_2]) \).
When a function is constant on cosets, there is an induced function \(\tilde{F}\) defined by \(\tilde{F}([n]) = F(n)\). Often, the tilde (~) symbol is omitted, which can lead to confusion.
**Task**: For each of the following functions \( F \), determine whether \( F \) is constant on cosets. Briefly justify your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecfe7738-95ba-450b-a609-248aa7606e96%2Fad07b330-4ff1-4048-b22b-c7a2c060b137%2Fx8qo1yv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Warmup for Next Week**
For each integer \( n \), define \([n] = n + 6\mathbb{Z}\), which represents the left coset of \( \mathbb{Z} \) by \( n \). Let \( \Omega = \{[0], [1], [2], [3], [4], [5]\} \) be the set of cosets.
A function \( F \), with domain \( \mathbb{Z} \), is considered **constant on cosets** if, for all \( n_1, n_2 \), if \([n_1] = [n_2]\), then \( F([n_1]) = F([n_2]) \).
When a function is constant on cosets, there is an induced function \(\tilde{F}\) defined by \(\tilde{F}([n]) = F(n)\). Often, the tilde (~) symbol is omitted, which can lead to confusion.
**Task**: For each of the following functions \( F \), determine whether \( F \) is constant on cosets. Briefly justify your answers.
Transcribed Image Text:The image contains a list of mathematical functions \( F(n) \) defined as follows:
(a) \( F(n) = n^3 \)
(b) \( F(n) = n \mod 2 \)
(c) \( F(n) = n \mod 3 \)
(d) \( F(n) = \) the largest prime factor \( n \) has.
(e) \( F(n) = 331 \)
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