Let A be a set and P(A) the power set of A. Complete the following proof that there is no onto function from A to P(A). Proof: Suppose (for contradiction) that f: A → P(A) is an onto function. Consider the set B = {r € A| x ¢ f(x)}. If f is onto, then for all C E P(A), there must be an element r E A so that f(x) = C. BC A so BE P(A), so there is an element x E A so that f(x) = B. %3D Question: is x E B or x 4 B?
Let A be a set and P(A) the power set of A. Complete the following proof that there is no onto function from A to P(A). Proof: Suppose (for contradiction) that f: A → P(A) is an onto function. Consider the set B = {r € A| x ¢ f(x)}. If f is onto, then for all C E P(A), there must be an element r E A so that f(x) = C. BC A so BE P(A), so there is an element x E A so that f(x) = B. %3D Question: is x E B or x 4 B?
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:3.
Let A be a set and P(A) the power set of A.
Complete the following proof that there is no onto function from A to P(A).
Proof: Suppose (for contradiction) that f : A→ P(A) is an onto function.
Consider the set B =
{x € A|x¢ f()}.
If f is onto, then for all C E P(A),
there must be an element r E A so that f(x) = C.
BCA so BE P(A), so there is an element x E A so that f(x) = B.
%3D
Question: is x E B or r 4 B?
1
4.
Prove that for all n E N, 7" +2 is divisible by 3.
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