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) A) Show that the set of all odd integers is countable, by producing a 1-1 unction F From J onto the set of all odd integers.

) A) Show that the set of all odd integers is countable, by producing a 1-1 unction F From J onto the set of all odd integers.

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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**Title: Countability of Sets and Functions** **1) A) Demonstrating the Countability of Odd Integers** - **Objective:** Show that the set of all odd integers is countable. - **Method:** Produce a one-to-one (1-1) correspondence function \( F \) from the set of natural numbers \( \mathbb{J} \) onto the set of all odd integers. **B) Establishing Countable Infinity of a Set** - **Given:** Suppose \( S \) is a set and there is a function \( F: \mathbb{J} \to S \) that is onto. - **Task:** Define a function \( g: S \to \mathbb{J} \) that is one-to-one (1-1). - **Conclusion:** Explain why establishing this function allows us to conclude that \( S \) is countably infinite.
Transcribed Image Text:**Title: Countability of Sets and Functions** **1) A) Demonstrating the Countability of Odd Integers** - **Objective:** Show that the set of all odd integers is countable. - **Method:** Produce a one-to-one (1-1) correspondence function \( F \) from the set of natural numbers \( \mathbb{J} \) onto the set of all odd integers. **B) Establishing Countable Infinity of a Set** - **Given:** Suppose \( S \) is a set and there is a function \( F: \mathbb{J} \to S \) that is onto. - **Task:** Define a function \( g: S \to \mathbb{J} \) that is one-to-one (1-1). - **Conclusion:** Explain why establishing this function allows us to conclude that \( S \) is countably infinite.
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