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LeA R be the relation defined the se4 Nx Z that (xy)EIC on so if and only if a) What is. the domain and Cange.

LeA R be the relation defined the se4 Nx Z that (xy)EIC on so if and only if a) What is. the domain and Cange.

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: Understanding Relations in Mathematics** **Content:** Let's explore the mathematical relation defined on the set \( \mathbb{N} \times \mathbb{Z} \). Here’s an explanation of the given problem and how to determine the domain and range. **Problem Statement:** Let \( R \) be the relation defined on the set \( \mathbb{N} \times \mathbb{Z} \) so that \((x, y) \in R\) if and only if \( x = y \). a.) What is the domain and range? **Explanation:** - **Relation \( R \):** The relation \( R \) is defined such that the ordered pair \((x, y)\) will be in \( R \) if \( x \) is equal to \( y \). This implies that the relation includes all pairs \((n, n)\) where \(n\) is a natural number that is also an integer. Therefore, the elements of the relation \( R \) are of the form \((n, n)\). - **Domain:** In this context, the domain is the set of all possible values of \( x \) that can appear in the relation. Since \( x \) can be any natural number that is also an integer, the domain of \( R \) is the set of all natural numbers, \( \mathbb{N} \). - **Range:** Similarly, the range is the set of all possible values of \( y \) that can appear in the relation. Because \( y \) must be equal to \( x \) and as \( x \) can each be any natural number, the range of \( R \) is also the set of all natural numbers, \( \mathbb{N} \). This exploration into relations reveals how elements are paired under certain conditions and helps to determine their domain and range in mathematical contexts.
Transcribed Image Text:**Title: Understanding Relations in Mathematics** **Content:** Let's explore the mathematical relation defined on the set \( \mathbb{N} \times \mathbb{Z} \). Here’s an explanation of the given problem and how to determine the domain and range. **Problem Statement:** Let \( R \) be the relation defined on the set \( \mathbb{N} \times \mathbb{Z} \) so that \((x, y) \in R\) if and only if \( x = y \). a.) What is the domain and range? **Explanation:** - **Relation \( R \):** The relation \( R \) is defined such that the ordered pair \((x, y)\) will be in \( R \) if \( x \) is equal to \( y \). This implies that the relation includes all pairs \((n, n)\) where \(n\) is a natural number that is also an integer. Therefore, the elements of the relation \( R \) are of the form \((n, n)\). - **Domain:** In this context, the domain is the set of all possible values of \( x \) that can appear in the relation. Since \( x \) can be any natural number that is also an integer, the domain of \( R \) is the set of all natural numbers, \( \mathbb{N} \). - **Range:** Similarly, the range is the set of all possible values of \( y \) that can appear in the relation. Because \( y \) must be equal to \( x \) and as \( x \) can each be any natural number, the range of \( R \) is also the set of all natural numbers, \( \mathbb{N} \). This exploration into relations reveals how elements are paired under certain conditions and helps to determine their domain and range in mathematical contexts.
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