Exercise 1. Let S(N) be the set of strictly increasing functions ƒ : N → N; ie. those functions of such that f(n) < f(m) for all 0 ≤ n < m. Prove that S(N) ≈ P(N). Exercise 2. A function f : N N is said to be a quasi-identity function if |f(n) - n ≤ 1 for all n = N. Determine whether the set QI(N) of quasi-identity functions is countable or uncountable. Exercise 3. Prove that if Sym(N) is the set of bijections : N → N, then card(Sym(N)) = 2. (Hint: consider the bijections : N → N such that [{ 2n, 2n+1}] = {2n, 2n+1} for all nЄ N.)
Exercise 1. Let S(N) be the set of strictly increasing functions ƒ : N → N; ie. those functions of such that f(n) < f(m) for all 0 ≤ n < m. Prove that S(N) ≈ P(N). Exercise 2. A function f : N N is said to be a quasi-identity function if |f(n) - n ≤ 1 for all n = N. Determine whether the set QI(N) of quasi-identity functions is countable or uncountable. Exercise 3. Prove that if Sym(N) is the set of bijections : N → N, then card(Sym(N)) = 2. (Hint: consider the bijections : N → N such that [{ 2n, 2n+1}] = {2n, 2n+1} for all nЄ N.)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
Related questions
Question
Set theory
![Exercise 1. Let S(N) be the set of strictly increasing functions ƒ : N → N;
ie. those functions of such that f(n) < f(m) for all 0 ≤ n < m.
Prove that S(N) ≈ P(N).
Exercise 2. A function f : N N is said to be a quasi-identity function if
|f(n) - n ≤ 1
for all n = N.
Determine whether the set QI(N) of quasi-identity functions is countable or
uncountable.
Exercise 3. Prove that if Sym(N) is the set of bijections : N → N,
then card(Sym(N)) = 2.
(Hint: consider the bijections : N → N such that [{ 2n, 2n+1}] = {2n, 2n+1}
for all nЄ N.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F79dc25b0-2b69-4ba8-928e-e4c10d38679c%2Fe1a4vpk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1. Let S(N) be the set of strictly increasing functions ƒ : N → N;
ie. those functions of such that f(n) < f(m) for all 0 ≤ n < m.
Prove that S(N) ≈ P(N).
Exercise 2. A function f : N N is said to be a quasi-identity function if
|f(n) - n ≤ 1
for all n = N.
Determine whether the set QI(N) of quasi-identity functions is countable or
uncountable.
Exercise 3. Prove that if Sym(N) is the set of bijections : N → N,
then card(Sym(N)) = 2.
(Hint: consider the bijections : N → N such that [{ 2n, 2n+1}] = {2n, 2n+1}
for all nЄ N.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 3 images
Recommended textbooks for you
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt