In this problem, we consider some functions that are based on the symmetries of the regular pentagon. b Let P = {a,b, c, d, e}, and let r : P → P be the function that rotates the pentagon clockwise by 72° (i.e., one fifth of a rotation). That is, r = {(a, b), (b, c), (c,d), (d, e), (e, a)}. Also, let fa : P → P be the function that reflects the pentagon along the line that passes through a and is perpendicular to cd. That is, fa= {(a, a), (b, e), (c, d), (d, c), (e, b)}. Now answer the following questions: (a) Is it true that fa or=rofa? Explain your answer. (b) Is it true that r−¹ = r (i.e., r is the inverse function of itself)? Explain your answer. (c) Is it true that f¹ = fa (i.e., fa is the inverse function of itself)? Explain your answer.
In this problem, we consider some functions that are based on the symmetries of the regular pentagon. b Let P = {a,b, c, d, e}, and let r : P → P be the function that rotates the pentagon clockwise by 72° (i.e., one fifth of a rotation). That is, r = {(a, b), (b, c), (c,d), (d, e), (e, a)}. Also, let fa : P → P be the function that reflects the pentagon along the line that passes through a and is perpendicular to cd. That is, fa= {(a, a), (b, e), (c, d), (d, c), (e, b)}. Now answer the following questions: (a) Is it true that fa or=rofa? Explain your answer. (b) Is it true that r−¹ = r (i.e., r is the inverse function of itself)? Explain your answer. (c) Is it true that f¹ = fa (i.e., fa is the inverse function of itself)? Explain 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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Transcribed Image Text:In this problem, we consider some functions that are based on the symmetries of the
regular pentagon.
b
Let P
{a, b, c, d, e}, and let r: P→ P be the function that rotates the pentagon clockwise
by 72° (i.e., one fifth of a rotation). That is,
r = {(a, b), (b, c), (c, d), (d, e), (e, a)}.
=
Also, let fa P → P be the function that reflects the pentagon along the line that passes
through a and is perpendicular to cd. That is,
fa = {(a, a), (b, e), (c, d), (d, c), (e, b)}.
Now answer the following questions:
(a) Is it true that fa or=ro fa? Explain your answer.
(b) Is it true that ri = r (i.e., r is the inverse function of itself)? Explain your answer.
(c) Is it true that fa¹ = fa (i.e., fa is the inverse function of itself)? Explain your answer.
(d) Let fo be the function that reflects the pentagon along the line that passes through b and
is perpendicular to de. Can you express f as a composition involving r and fa? Explain
why the composition you propose works.
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