1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A) and BC P(B), and let f : A →→ B. Define f(A) = {f(X): X € A} and f-¹(B) = {f¹(Y): Y EB}. (a) If B is a o-algebra, prove that f-¹(B) is also a o-algebra. (b) Give an example to show that if A is a o-algebra, f(A) need not be a o-algebra.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A)
=
{f(X) XE A} and
and B C P(B), and let f : A → B. Define f(A)
f¹(B) = {f¹(Y) : Y = B}.
(a) If B is a o-algebra, prove that f-¹(B) is also a o-algebra.
(b) Give an example to show that if A is a o-algebra, f(A) need not be a
o-algebra.
Transcribed Image Text:1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A) = {f(X) XE A} and and B C P(B), and let f : A → B. Define f(A) f¹(B) = {f¹(Y) : Y = B}. (a) If B is a o-algebra, prove that f-¹(B) is also a o-algebra. (b) Give an example to show that if A is a o-algebra, f(A) need not be a o-algebra.
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