- Find a bijection between Z and E (the even numbers), prove your answer is correct.
- Find a bijection between Z and E (the even numbers), prove your answer is correct.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer just question 5,6,7,8
![mark). In some cases you may feel comfortable using "prose" proofs, but make certain your
conclusion is either the last line of a table-based proof or stated as a conclusion so the marke
does not have to guess.
Assignment 6 Functions and countability
1. Suppose f(x) = 2x +3, g(x) = 17 – x', and h(x) = (x + 1). Recall the definition of
function composition p o q, and show functions resulting from the composition of
fog,f •g•h,h •f.
%3D
2. Consider f(x) = 2x + 3. Is f: N N one-to-one? Is it onto? Prove, or give a
counterexample.
3. Consider f(x) = 5+ 4x is a bijection using the definition in class. Is f : Z → Z one-to-
one? Is it onto?
4. Let S = {xEN|3z(z E NAZ = x)}. Letf : N → S be the function f (x) = x². Is f
1-1? Onto? Show why.
5. Find a bijection between Z and E (the even numbers), prove your answer is correct.
6. Consider the infinite set W of all finite strings of binary numerals 0, 1, for example:
10
11
100
Demonstrate whether W is countable set.
7. Consider the infinite set Yof all infinite strings of binary numbers 0,1, for example
011011100010101...
000001010101001...
Show whether Y is a countable set, and show why.
8. Bonus question. Find a bijection between [0,1] and [0,1) (i.e., from the closed interval to the
half open interval.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3998581a-2a2a-4b7c-8517-697c99ec7fd7%2F477311ba-0022-4f9b-a074-6705e52adf0b%2Fvil4jzo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:mark). In some cases you may feel comfortable using "prose" proofs, but make certain your
conclusion is either the last line of a table-based proof or stated as a conclusion so the marke
does not have to guess.
Assignment 6 Functions and countability
1. Suppose f(x) = 2x +3, g(x) = 17 – x', and h(x) = (x + 1). Recall the definition of
function composition p o q, and show functions resulting from the composition of
fog,f •g•h,h •f.
%3D
2. Consider f(x) = 2x + 3. Is f: N N one-to-one? Is it onto? Prove, or give a
counterexample.
3. Consider f(x) = 5+ 4x is a bijection using the definition in class. Is f : Z → Z one-to-
one? Is it onto?
4. Let S = {xEN|3z(z E NAZ = x)}. Letf : N → S be the function f (x) = x². Is f
1-1? Onto? Show why.
5. Find a bijection between Z and E (the even numbers), prove your answer is correct.
6. Consider the infinite set W of all finite strings of binary numerals 0, 1, for example:
10
11
100
Demonstrate whether W is countable set.
7. Consider the infinite set Yof all infinite strings of binary numbers 0,1, for example
011011100010101...
000001010101001...
Show whether Y is a countable set, and show why.
8. Bonus question. Find a bijection between [0,1] and [0,1) (i.e., from the closed interval to the
half open interval.)
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