Exercise 17.3.11. Let f: {-3,-2,-1,0, 1,2,3} → Z be defined by f(x) = x². (a) What is the range of f? (b) For every number n in the range of f, find the set of all numbers in the domain of f that map to n. Denote this set as An (for example, if we let n = 0, then only 0 maps to 0, so Ao = 0). List the elements of An for each n in the range of f. (c) Show that the sets {An} that you listed in part (b) form a partition of the domain of f. (d) According to Proposition 17.4.11, this partition produces an equivalence relation on the domain of f. Draw a digraph that represents the equiv- alence relation. (e) We also know that the function f produces an equivalence relation on the domain of f, as in Proposition 17.3.6. Draw a digraph that represents this equivalence relation. (f) What may you conclude from your results in (d) and (e)?
Exercise 17.3.11. Let f: {-3,-2,-1,0, 1,2,3} → Z be defined by f(x) = x². (a) What is the range of f? (b) For every number n in the range of f, find the set of all numbers in the domain of f that map to n. Denote this set as An (for example, if we let n = 0, then only 0 maps to 0, so Ao = 0). List the elements of An for each n in the range of f. (c) Show that the sets {An} that you listed in part (b) form a partition of the domain of f. (d) According to Proposition 17.4.11, this partition produces an equivalence relation on the domain of f. Draw a digraph that represents the equiv- alence relation. (e) We also know that the function f produces an equivalence relation on the domain of f, as in Proposition 17.3.6. Draw a digraph that represents this equivalence relation. (f) What may you conclude from your results in (d) and (e)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 17.3.11 part ABCDE. Please show step by step and explain
Transcribed Image Text:Exercise 17.3.11. Let f: {-3,-2,-1,0, 1,2,3} → Z be defined by f(x) =
x².
(a) What is the range of f?
(b) For every number n in the range of f, find the set of all numbers in the
domain of f that map to n. Denote this set as An (for example, if we
let n = 0, then only 0 maps to 0, so Ao = 0). List the elements of An
for each n in the range of f.
(c) Show that the sets {An} that you listed in part (b) form a partition of
the domain of f.
(d) According to Proposition 17.4.11, this partition produces an equivalence
relation on the domain of f. Draw a digraph that represents the equiv-
alence relation.
(e) We also know that the function f produces an equivalence relation on the
domain of f, as in Proposition 17.3.6. Draw a digraph that represents
this equivalence relation.
(f) What may you conclude from your results in (d) and (e)?
![Proposition 17.3.6. Suppose f: A→ B. If we define a binary relation
on A by
f(a₁) = f(a₂),
a1 ~ az
then~ is an equivalence relation on A.
Proposition 17.4.11.
Then
is a partition of A.
Suppose ~ is an equivalence relation on a set A.
{[a] | a € A}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F0a7a306a-8141-4a8f-a755-17d427fe622f%2Ftthfhb7_processed.png&w=3840&q=75)
Transcribed Image Text:Proposition 17.3.6. Suppose f: A→ B. If we define a binary relation
on A by
f(a₁) = f(a₂),
a1 ~ az
then~ is an equivalence relation on A.
Proposition 17.4.11.
Then
is a partition of A.
Suppose ~ is an equivalence relation on a set A.
{[a] | a € A}
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