Proof Let f: X Y be an arbitraryfunction and F≤ P(Y). Show that,o(f¹(F)) = f-¹(o(F))..

Answered: Let f: XY be an arbitrary function and…

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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Related questions

Question

Proof

Let f: X Y be an arbitrary
function and F≤ P(Y). Show that,
o(f¹(F)) = f-¹(o(F))..

Expert Solution

Step 1

What is Sigma-Algebra:

A $σ$ -algebra (also $σ$ -field) on a set X is a nonempty collection of X subsets closed under complement, countable unions, and countable crossings in mathematical analysis and probability theory. X and are referred to as a measurable space. The set algebras, which the $σ$ -algebras are a subset of, merely require that their elements be closed under the union or intersection of finitely many subsets, a lesser requirement.

Given:

Given that $f : X \to Y$ is an arbitrary function and $F \subseteq P (Y)$ .

To Prove:

We prove that $f^{- 1} (σ (F)) = σ (f^{- 1} (F))$ .