[Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that S and T are disjoint (i.e., SnT = Ø) if and only if the following condition holds: (*) For all subsets A₁, A₂ C S and B₁, B₂ C T, if A₁ U B₁ = A₂ U B2, then A₁ = A2 and B₁ B2₂. =
[Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that S and T are disjoint (i.e., SnT = Ø) if and only if the following condition holds: (*) For all subsets A₁, A₂ C S and B₁, B₂ C T, if A₁ U B₁ = A₂ U B2, then A₁ = A2 and B₁ B2₂. =
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. [Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that
S and T are disjoint (i.e., SnT = Ø) if and only if the following condition holds:
(*)
For all subsets A₁, A₂ C S and B₁, B2 C T, if A₁ U B₁ = A₂ U B2,
then A₁
A2 and B₁ B₂.
Note: The condition (+) is equivalent to the statement that the function
=
=
ƒ: P(S) × P(T) → P(SŪT)
(A, B) → AUB
is injective. You proved on Homework 22 that this function is always surjective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7626d59-6405-4948-a625-19aca32c9eb7%2F5feaddfa-ab63-4c4e-a0be-09fb42e5dab8%2F7no5zcq_processed.png&w=3840&q=75)
Transcribed Image Text:1. [Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that
S and T are disjoint (i.e., SnT = Ø) if and only if the following condition holds:
(*)
For all subsets A₁, A₂ C S and B₁, B2 C T, if A₁ U B₁ = A₂ U B2,
then A₁
A2 and B₁ B₂.
Note: The condition (+) is equivalent to the statement that the function
=
=
ƒ: P(S) × P(T) → P(SŪT)
(A, B) → AUB
is injective. You proved on Homework 22 that this function is always surjective.
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