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Let A, B, C be nonempty sets and let f : AB, g: B → C be functions such that go f is bijective. Prove that g is injective if and only if f is surjective.

Let A, B, C be nonempty sets and let f : AB, g: B → C be functions such that go f is bijective. Prove that g is injective if and only if f is surjective.

Advanced Engineering Mathematics
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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**Mathematical Problem: Injective and Surjective Functions** **Problem Statement:** Let \( A, B, C \) be nonempty sets and let \( f: A \to B \), \( g: B \to C \) be functions such that the composition \( g \circ f \) is bijective. Prove that \( g \) is injective if and only if \( f \) is surjective. **Explanation:** The problem explores the conditions under which the composition of two functions is both injective (one-to-one) and surjective (onto), thereby making it bijective. It focuses on: - **Surjectivity (Onto):** Every element of the set \( B \) is an image of at least one element from the set \( A \). - **Injectivity (One-to-One):** Each element of the set \( B \) is mapped to a unique element in \( A \). - **Bijectivity:** A function that is both injective and surjective. The aim is to demonstrate a relationship between the injectivity of \( g \) and the surjectivity of \( f \) given that their composition \( g \circ f \) is bijective.
Transcribed Image Text:**Mathematical Problem: Injective and Surjective Functions** **Problem Statement:** Let \( A, B, C \) be nonempty sets and let \( f: A \to B \), \( g: B \to C \) be functions such that the composition \( g \circ f \) is bijective. Prove that \( g \) is injective if and only if \( f \) is surjective. **Explanation:** The problem explores the conditions under which the composition of two functions is both injective (one-to-one) and surjective (onto), thereby making it bijective. It focuses on: - **Surjectivity (Onto):** Every element of the set \( B \) is an image of at least one element from the set \( A \). - **Injectivity (One-to-One):** Each element of the set \( B \) is mapped to a unique element in \( A \). - **Bijectivity:** A function that is both injective and surjective. The aim is to demonstrate a relationship between the injectivity of \( g \) and the surjectivity of \( f \) given that their composition \( g \circ f \) is bijective.
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