Problem five : Find all maps : (Z, +) → (Q*, x) satisfying O (a + b) = 6(a) x o(b) Va, be Z
Problem five : Find all maps : (Z, +) → (Q*, x) satisfying O (a + b) = 6(a) x o(b) Va, be Z
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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![### Problem Five:
**Objective:**
Find all maps \( \phi : (\mathbb{Z}, +) \rightarrow (\mathbb{Q}^*, \times) \) satisfying the following condition:
\[ \phi(a + b) = \phi(a) \times \phi(b) \quad \forall \, a, b \in \mathbb{Z} \]
where \( \mathbb{Z} \) represents the set of all integers under addition, and \( \mathbb{Q}^* \) represents the set of all non-zero rational numbers under multiplication.
### Explanation:
The problem is asking to determine all functions \( \phi \) that map from the set of integers \( \mathbb{Z} \) to the set of non-zero rational numbers \( \mathbb{Q}^* \), such that the function \( \phi \) converts the addition operation in \( \mathbb{Z} \) to the multiplication operation in \( \mathbb{Q}^* \). This means that for any integers \( a \) and \( b \), the function \( \phi \) must satisfy:
\[ \phi(a + b) = \phi(a) \times \phi(b) \]
### Steps to Find \( \phi \):
1. **Initial Conditions:** Identify the value of \( \phi \) at specific points, such as \( \phi(0) \) and \( \phi(1) \).
2. **Functional Equation:** Use the given functional equation to determine the values of \( \phi \) at other integers by leveraging properties of integer addition.
3. **Verify Properties:** Check if the discovered function meets all required properties of the mapping from \( \mathbb{Z} \) to \( \mathbb{Q}^* \).
This process involves mathematical reasoning and possibly solving algebraic equations to determine the form of the function \( \phi \).
### Note:
This problem is classic in the study of homomorphisms between algebraic structures, specifically between the group of integers under addition and the group of non-zero rational numbers under multiplication. Homomorphisms of this sort are essential in understanding the structure and behavior of algebraic systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5515fa1-70cf-4ea0-b2f5-c182d91d6664%2F463b2b48-c116-4749-bb07-dd1cc1e42b19%2F55mswap_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Five:
**Objective:**
Find all maps \( \phi : (\mathbb{Z}, +) \rightarrow (\mathbb{Q}^*, \times) \) satisfying the following condition:
\[ \phi(a + b) = \phi(a) \times \phi(b) \quad \forall \, a, b \in \mathbb{Z} \]
where \( \mathbb{Z} \) represents the set of all integers under addition, and \( \mathbb{Q}^* \) represents the set of all non-zero rational numbers under multiplication.
### Explanation:
The problem is asking to determine all functions \( \phi \) that map from the set of integers \( \mathbb{Z} \) to the set of non-zero rational numbers \( \mathbb{Q}^* \), such that the function \( \phi \) converts the addition operation in \( \mathbb{Z} \) to the multiplication operation in \( \mathbb{Q}^* \). This means that for any integers \( a \) and \( b \), the function \( \phi \) must satisfy:
\[ \phi(a + b) = \phi(a) \times \phi(b) \]
### Steps to Find \( \phi \):
1. **Initial Conditions:** Identify the value of \( \phi \) at specific points, such as \( \phi(0) \) and \( \phi(1) \).
2. **Functional Equation:** Use the given functional equation to determine the values of \( \phi \) at other integers by leveraging properties of integer addition.
3. **Verify Properties:** Check if the discovered function meets all required properties of the mapping from \( \mathbb{Z} \) to \( \mathbb{Q}^* \).
This process involves mathematical reasoning and possibly solving algebraic equations to determine the form of the function \( \phi \).
### Note:
This problem is classic in the study of homomorphisms between algebraic structures, specifically between the group of integers under addition and the group of non-zero rational numbers under multiplication. Homomorphisms of this sort are essential in understanding the structure and behavior of algebraic systems.
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