Answered: Let H 5 {0, ±3, ±6, ±9, . . .}. Find…

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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Question

Let H 5 {0, ±3, ±6, ±9, . . .}. Find all the left cosets of H in Z.

Expert Solution

Step 1

The given subset $H = \{0, \pm 3, \pm 6, \pm 9, . . . .\}$ .

We have to find all the left cosets of $H$ in $ℤ$ .

Step 2

Left coset:

If $G$ be any group and $H$ is any nonempty subset of $G$ .

The left-coset of $H$ is $a H = \{a h | h \in H\}$ , for any $a \in G$ .

Now,

$\begin{array}{rcl} H & = & \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ = & 3 \{0, \pm 1, \pm 2, \pm 3, . . . .\} \\ = & 3 ℤ \\ H & = & \{3 k | k \in ℤ\} \end{array}$

The classes of $n ℤ$ are $\{ℤ, ℤ + 1, ℤ + 2, . . . . . ., ℤ + (n - 1)\}$ .

From this $\{3 k | k \in ℤ\}, \{3 k + 1 | k \in ℤ\}, \{3 k + 2 | k \in ℤ\}$

For $x = 3 k, k \in ℤ$ :

$\begin{array}{rcl} x + H & = & 3 k + \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ 3 k & \in & \{0, \pm 3, \pm 6, \pm 9, . . . .\} = H \\ x + H & = & H \end{array}$

i.e., $\{a H = H i f a \in H\}$

For $x = 3 k + 1, k \in ℤ$ :

$\begin{array}{rcl} x + H & = & 3 k + 1 + \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ x + H & = & 1 + \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ x + H & = & 1 + H \end{array}$

For $x = 3 k + 2, k \in ℤ$ :

$\begin{array}{rcl} x + H & = & 3 k + 2 + \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ x + H & = & 2 + \{0, \pm 3, \pm 6, \pm 9, . . . .\} \\ x + H & = & 2 + H \end{array}$

Therefore, the left coset of $H$ in $ℤ$ are $\{H, 1 + H, 2 + H\}$ .

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