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For this one can you please explain to me How does number 1 is differ from a relation?

For this one can you please explain to me How does number 1 is differ from a relation?

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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For this one can you please explain to me How does number 1 is differ from a relation? I’m trying to implication the first one. Please help me thanks
**Functions** In Section 1.2, we showed that ordered pairs can be defined in terms of sets and we defined Cartesian products in terms of ordered pairs. In this section, we introduced relations as subsets of Cartesian products. Thus, we can now define functions in a way that depends only on the concept of set. Although this definition is not obviously related to the way we usually work with functions in mathematics, it is satisfying from a theoretical point of view, and computer scientists like it because it is particularly well suited for operating with functions on a computer. **Definition** A **function** \( F \) from a set \( A \) to a set \( B \) is a relation with domain \( A \) and co-domain \( B \) that satisfies the following two properties: 1. For every element \( x \) in \( A \), there is an element \( y \) in \( B \) such that \( (x, y) \in F \). 2. For all elements \( x \) in \( A \) and \( y \) and \( z \) in \( B \), if \( (x, y) \in F \) and \( (x, z) \in F \), then \( y = z \).
Transcribed Image Text:**Functions** In Section 1.2, we showed that ordered pairs can be defined in terms of sets and we defined Cartesian products in terms of ordered pairs. In this section, we introduced relations as subsets of Cartesian products. Thus, we can now define functions in a way that depends only on the concept of set. Although this definition is not obviously related to the way we usually work with functions in mathematics, it is satisfying from a theoretical point of view, and computer scientists like it because it is particularly well suited for operating with functions on a computer. **Definition** A **function** \( F \) from a set \( A \) to a set \( B \) is a relation with domain \( A \) and co-domain \( B \) that satisfies the following two properties: 1. For every element \( x \) in \( A \), there is an element \( y \) in \( B \) such that \( (x, y) \in F \). 2. For all elements \( x \) in \( A \) and \( y \) and \( z \) in \( B \), if \( (x, y) \in F \) and \( (x, z) \in F \), then \( y = z \).
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