16. Prove using the First Principle of Math is divisible by 4 for all integers n ≥ 1 6.7-2.3" Induction (weak induction):
16. Prove using the First Principle of Math is divisible by 4 for all integers n ≥ 1 6.7-2.3" Induction (weak induction):
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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### Advanced Mathematics Problems
#### 16. Mathematical Induction Problem
Prove using the First Principle of Mathematical Induction (weak induction):
\[ 6 \cdot 7^n - 2 \cdot 3^n \text{ is divisible by 4 for all integers } n \geq 1. \]
#### 17. Relations on the Set of Natural Numbers
Let relation \( G \) be a subset of the cross product of the natural numbers with the natural numbers: \( G \subseteq \mathbb{N} \times \mathbb{N} \). Define relation \( R \) as:
\[ (a, b) \in G \Rightarrow [a + b \geq 18] \]
You may assume the natural numbers include 0. Determine if the relation is each of these:
- (a) Reflexive: **YES**
- (b) Symmetric: **YES**
- (c) Transitive: **YES**
- (d) Antisymmetric: **NO**
- (e) Irreflexive: **YES**
- (f) Asymmetric: **NO**
#### Explanation of Terms:
- **Reflexive**: A relation \( R \) on a set \( A \) is reflexive if every element of \( A \) is related to itself.
- **Symmetric**: A relation \( R \) on a set \( A \) is symmetric if for all \( a \) and \( b \) in \( A \), if \( a \) is related to \( b \), then \( b \) is related to \( a \).
- **Transitive**: A relation \( R \) on a set \( A \) is transitive if for all \( a, b, \) and \( c \) in \( A \), if \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \).
- **Antisymmetric**: A relation \( R \) on a set \( A \) is antisymmetric if for all \( a \) and \( b \) in \( A \), if \( a \) is related to \( b \) and \( b \) is related to \( a \), then \( a = b \).
- **Irreflexive**: A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb017a202-bb10-4722-a8c7-7967a12b195b%2F5c90b01a-f9a9-400c-9d0a-cc6a01a023a7%2F3pshxve_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Advanced Mathematics Problems
#### 16. Mathematical Induction Problem
Prove using the First Principle of Mathematical Induction (weak induction):
\[ 6 \cdot 7^n - 2 \cdot 3^n \text{ is divisible by 4 for all integers } n \geq 1. \]
#### 17. Relations on the Set of Natural Numbers
Let relation \( G \) be a subset of the cross product of the natural numbers with the natural numbers: \( G \subseteq \mathbb{N} \times \mathbb{N} \). Define relation \( R \) as:
\[ (a, b) \in G \Rightarrow [a + b \geq 18] \]
You may assume the natural numbers include 0. Determine if the relation is each of these:
- (a) Reflexive: **YES**
- (b) Symmetric: **YES**
- (c) Transitive: **YES**
- (d) Antisymmetric: **NO**
- (e) Irreflexive: **YES**
- (f) Asymmetric: **NO**
#### Explanation of Terms:
- **Reflexive**: A relation \( R \) on a set \( A \) is reflexive if every element of \( A \) is related to itself.
- **Symmetric**: A relation \( R \) on a set \( A \) is symmetric if for all \( a \) and \( b \) in \( A \), if \( a \) is related to \( b \), then \( b \) is related to \( a \).
- **Transitive**: A relation \( R \) on a set \( A \) is transitive if for all \( a, b, \) and \( c \) in \( A \), if \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \).
- **Antisymmetric**: A relation \( R \) on a set \( A \) is antisymmetric if for all \( a \) and \( b \) in \( A \), if \( a \) is related to \( b \) and \( b \) is related to \( a \), then \( a = b \).
- **Irreflexive**: A
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