(a) Having just finished constructing his time machine, Kik has to adjust one last setting before using his machine. He needs to choose 4 chronologies in time, represented by integers p, q, r, s such that p≤q≤r≤s. Any user of the machine will be able to travel from a year in the interval [p..r] to a year in the interval [q..s]. Help Kik analyse the results of his first trial by determining how many distinct functions there are with domain [p..r], and codomain [q..s].
(a) Having just finished constructing his time machine, Kik has to adjust one last setting before using his machine. He needs to choose 4 chronologies in time, represented by integers p, q, r, s such that p≤q≤r≤s. Any user of the machine will be able to travel from a year in the interval [p..r] to a year in the interval [q..s]. Help Kik analyse the results of his first trial by determining how many distinct functions there are with domain [p..r], and codomain [q..s].
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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![(a) Having just finished constructing his time machine, Kik has to adjust one last setting before
using his machine. He needs to choose 4 chronologies in time, represented by integers
p, q, r, s such that p≤ q ≤r ≤s. Any user of the machine will be able to travel from a
year in the interval [p..r] to a year in the interval [q..s]. Help Kik analyse the results of
his first trial by determining how many distinct functions there are with domain [p..r], and
codomain [q..s].
(b) Kik is not satisfied with the results of the first trial. In order to make improvements, Kik
decides to run a second trial focusing on a specific form of functions. He is interested in
functions of the form f: [1..n] → [1..2n], where n is a positive integer, such that for all
in the domain we have either f(x) < x or f(x) ≥ 2x. How many distinct functions of that
form exist? Justify your answer and put it in closed form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac0fe233-9215-469b-9b52-607d6dc120fe%2Fbd173cbc-87f5-419b-8a6f-95898dad1071%2F6yo80jk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Having just finished constructing his time machine, Kik has to adjust one last setting before
using his machine. He needs to choose 4 chronologies in time, represented by integers
p, q, r, s such that p≤ q ≤r ≤s. Any user of the machine will be able to travel from a
year in the interval [p..r] to a year in the interval [q..s]. Help Kik analyse the results of
his first trial by determining how many distinct functions there are with domain [p..r], and
codomain [q..s].
(b) Kik is not satisfied with the results of the first trial. In order to make improvements, Kik
decides to run a second trial focusing on a specific form of functions. He is interested in
functions of the form f: [1..n] → [1..2n], where n is a positive integer, such that for all
in the domain we have either f(x) < x or f(x) ≥ 2x. How many distinct functions of that
form exist? Justify your answer and put it in closed form.
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