**Problem Statement:**Let \( f: X \to Y \) be a function. For any subset \( A \subseteq Y \), let \( f^{-1}(A) \) be the pre-image of \( A \) in \( X \). Show that this defines a function from \(\mathcal{P}(Y)\) to \(\mathcal{P}(X)\). Can you say when it is injective or surjective?**Explanation:**This mathematical problem deals with the concept of pre-images and power sets. - **Power Set, \(\mathcal{P}(Y)\):** The power set of a set \( Y \) is the set of all subsets of \( Y \).- **Pre-image, \( f^{-1}(A) \):** The pre-image of \( A \) under \( f \) is the set of all elements in \( X \) whose image under \( f \) is in \( A \).The task is to show that for a function \( f: X \to Y \), this mapping from subsets of \( Y \) to subsets of \( X \) indeed constitutes a function from \(\mathcal{P}(Y)\) to \(\mathcal{P}(X)\). Additionally, the problem asks to explore the injectivity and surjectivity of this induced function. In terms of injectivity, the function is injective if different subsets of \( Y \) have different pre-images in \( X \). In terms of surjectivity, for every subset in \(\mathcal{P}(X)\), there exists a subset in \(\mathcal{P}(Y)\) such that its pre-image is the given subset in \( X \).**Related Concepts:**- **Injective Function (One-to-One):** A function is injective if it maps distinct elements in the domain to distinct elements in the codomain.- **Surjective Function (Onto):** A function is surjective if every element in the codomain is an image of at least one element in the domain.This problem involves analyzing these properties to understand the behavior of the pre-image function transforming subsets of the codomain to subsets of the domain.

Answered: Image /qna-images/answer/a2c0cb30-b22a-4fa3-a863-532aefb0d531.jpg

Answered: et f: X → Y be a function. For any…

et f: X → Y be a function. For any subset A C Y, let ƒ-¹(A) be the re-image of A in X. Show that this defines a function from P(Y) to P(X). Can you say when it is injective or surjective?

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

**Problem Statement:**

Let \( f: X \to Y \) be a function. For any subset \( A \subseteq Y \), let \( f^{-1}(A) \) be the pre-image of \( A \) in \( X \). Show that this defines a function from \(\mathcal{P}(Y)\) to \(\mathcal{P}(X)\). Can you say when it is injective or surjective?

**Explanation:**

This mathematical problem deals with the concept of pre-images and power sets.

- **Power Set, \(\mathcal{P}(Y)\):** The power set of a set \( Y \) is the set of all subsets of \( Y \).
- **Pre-image, \( f^{-1}(A) \):** The pre-image of \( A \) under \( f \) is the set of all elements in \( X \) whose image under \( f \) is in \( A \).

The task is to show that for a function \( f: X \to Y \), this mapping from subsets of \( Y \) to subsets of \( X \) indeed constitutes a function from \(\mathcal{P}(Y)\) to \(\mathcal{P}(X)\). Additionally, the problem asks to explore the injectivity and surjectivity of this induced function.

In terms of injectivity, the function is injective if different subsets of \( Y \) have different pre-images in \( X \). In terms of surjectivity, for every subset in \(\mathcal{P}(X)\), there exists a subset in \(\mathcal{P}(Y)\) such that its pre-image is the given subset in \( X \).

**Related Concepts:**

- **Injective Function (One-to-One):** A function is injective if it maps distinct elements in the domain to distinct elements in the codomain.
- **Surjective Function (Onto):** A function is surjective if every element in the codomain is an image of at least one element in the domain.

This problem involves analyzing these properties to understand the behavior of the pre-image function transforming subsets of the codomain to subsets of the domain.

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