1. Write the following permutations in cycle notation. (1bd from 5.4) 23 4 5 2 3 4 5 a) (1 b) (1 4 25 1 3. 4 3 2 5.
1. Write the following permutations in cycle notation. (1bd from 5.4) 23 4 5 2 3 4 5 a) (1 b) (1 4 25 1 3. 4 3 2 5.
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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![**Permutation and Cycle Notation in Mathematics**
### Problem Statement:
1. Write the following permutations in cycle notation. (1bd from 5.4)
#### a)
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{matrix}
\right)
\]
#### b)
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{matrix}
\right)
\]
### Explanation:
In permutation cycle notation, we rearrange the elements of a given set into cycles that showcase the mapping from one element to another. Each cycle in the notation captures the transition from one element to the next in sequence, ending back at the starting element.
#### Detailed Transformation:
For part (a):
**Original Permutation:**
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{matrix}
\right)
\]
**Cycle Notation:**
- Start with 1: 1 maps to 4, and 4 maps back to 1. Hence, we have (1 4).
- Element 2 maps to 2, so it's a single element cycle: (2).
- Then start with 3: 3 maps to 5, and 5 maps back to 3. Hence, (3 5).
Combining all cycles, the compact cycle notation for part (a) is:
\[
(1 \ 4)(2)(3 \ 5)
\]
For part (b):
**Original Permutation:**
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{matrix}
\right)
\]
**Cycle Notation:**
- Element 1 maps to itself: (1).
- Start with 2: 2 maps to 4, and 4 maps to 2. Hence, (2 4).
- Element 3 maps to itself: (3).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed14a3ea-da26-4be7-a143-8b845df95e91%2Ff810503d-e621-436b-82b8-e642b629b6ef%2Fov88t5e_processed.png&w=3840&q=75)
Transcribed Image Text:**Permutation and Cycle Notation in Mathematics**
### Problem Statement:
1. Write the following permutations in cycle notation. (1bd from 5.4)
#### a)
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{matrix}
\right)
\]
#### b)
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{matrix}
\right)
\]
### Explanation:
In permutation cycle notation, we rearrange the elements of a given set into cycles that showcase the mapping from one element to another. Each cycle in the notation captures the transition from one element to the next in sequence, ending back at the starting element.
#### Detailed Transformation:
For part (a):
**Original Permutation:**
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{matrix}
\right)
\]
**Cycle Notation:**
- Start with 1: 1 maps to 4, and 4 maps back to 1. Hence, we have (1 4).
- Element 2 maps to 2, so it's a single element cycle: (2).
- Then start with 3: 3 maps to 5, and 5 maps back to 3. Hence, (3 5).
Combining all cycles, the compact cycle notation for part (a) is:
\[
(1 \ 4)(2)(3 \ 5)
\]
For part (b):
**Original Permutation:**
\[
\left(
\begin{matrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{matrix}
\right)
\]
**Cycle Notation:**
- Element 1 maps to itself: (1).
- Start with 2: 2 maps to 4, and 4 maps to 2. Hence, (2 4).
- Element 3 maps to itself: (3).
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