1. We'll need to establish the Schröder-Bernstein Theorem first. Assume there exists a 1 – 1 function f:X→Y and another function g : Y → X. Follow the steps to show that there exists a 1 – 1 and onto function h : X → Y and hence X ~ Y.
1. We'll need to establish the Schröder-Bernstein Theorem first. Assume there exists a 1 – 1 function f:X→Y and another function g : Y → X. Follow the steps to show that there exists a 1 – 1 and onto function h : X → Y and hence X ~ Y.
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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Question
![Write a rigorous proof for the following statement.
Statement: P(N)~R
follow the road map I have provided below, make sure the proof is
logically sound and that the logic is clear to the reader (not just you)
1 function f :X → Y and
1. We'll need to establish the Schröder-Bernstein Theorem first. Assume there exists a 1 –
another function g : Y → X. Follow the steps to show that there exists a 1 – 1 and onto function h : X → Y and hence
X - Y.
1.1. The strategy is to partition X and Y into components
X = AU A'
with ANA' = Ø and BNB' = Ø, in such a way that f maps A onto B, and g maps B' onto A'. Explain how this
would lead to a proof that X ~ Y.
and
Y = BUB'
1.2. Set Aj = X\g(Y) = {x EX|x¢ g(Y)} (What happens if A1 = Ø?) and inductively define a sequence of sets by
letting An+1 = g(S(A,)). Show that {A» |n EN} is a pairwise disjoint collection of subsets of X, while
{F(An) |n EN} is a similar collection in Y.
%3|
1.3. Let A = U14n and B = Uiƒ(An). Show that f maps A onto B.
%3D
1.4. Let A' =X\A and B' = Y\B. Show that
g maps B' onto A'.
2. Find a 1 -
1 function f : P(N) → R. Be careful not to break mathematics by finding a 1 – 1 function f : P(N) → N!
3. Find a 1 – 1 function f : R → P(N) and conclude that P(N) ~ R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87bd9bd0-40fd-4172-a50a-abb52eb6a8c1%2F8fdc58dd-44fe-48ce-9ef9-aca34b8e94ca%2Fpy11xfk_processed.png&w=3840&q=75)
Transcribed Image Text:Write a rigorous proof for the following statement.
Statement: P(N)~R
follow the road map I have provided below, make sure the proof is
logically sound and that the logic is clear to the reader (not just you)
1 function f :X → Y and
1. We'll need to establish the Schröder-Bernstein Theorem first. Assume there exists a 1 –
another function g : Y → X. Follow the steps to show that there exists a 1 – 1 and onto function h : X → Y and hence
X - Y.
1.1. The strategy is to partition X and Y into components
X = AU A'
with ANA' = Ø and BNB' = Ø, in such a way that f maps A onto B, and g maps B' onto A'. Explain how this
would lead to a proof that X ~ Y.
and
Y = BUB'
1.2. Set Aj = X\g(Y) = {x EX|x¢ g(Y)} (What happens if A1 = Ø?) and inductively define a sequence of sets by
letting An+1 = g(S(A,)). Show that {A» |n EN} is a pairwise disjoint collection of subsets of X, while
{F(An) |n EN} is a similar collection in Y.
%3|
1.3. Let A = U14n and B = Uiƒ(An). Show that f maps A onto B.
%3D
1.4. Let A' =X\A and B' = Y\B. Show that
g maps B' onto A'.
2. Find a 1 -
1 function f : P(N) → R. Be careful not to break mathematics by finding a 1 – 1 function f : P(N) → N!
3. Find a 1 – 1 function f : R → P(N) and conclude that P(N) ~ R.
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