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

Question

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Answer 1

Textbook Question

Chapter 7, Problem 1E

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

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

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Answer 2

Textbook Question

Chapter 7, Problem 1E

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

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

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Answer 3

Textbook Question

Chapter 7, Problem 1E

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

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

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Answer 4

Textbook Question

Chapter 7, Problem 1E

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

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Answer 8

Expert Solution & Answer

To determine

To find:

The all left costs of H in Z of the given sets.

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To find:

The all left costs of H in Z of the given sets.

Answer 12

Answer to Problem 1E

H,1+H,2+H

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Answer 13

Explanation of Solution

Given:

H={0,±3,±6,±9,.......}

Concept used:

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

The left-coset of H is aH={ah|h∈H} For any a∈G .

Calculation:

H={0,±3,±6,±9,.......}H=3{0,±1,±2,±3,.......}H=3ZH={3k|k∈Z}

The classes of nZ

{Z,Z+1,Z+2,......Z+(n−1)}

Therefore, the classes of 3Z are {Z,Z+1,Z+2} .

That is

{3k|k∈Z},{3k+1|k∈Z},{3k+2|k∈Z}

Let

x=3k,k∈Zx+H=3k+{0,±3,±6,±9,.......}3k∈{0,±3,±6,±9,.......}=Hx+H=3k

Where

{aH=H if a∈H}

Let

x=3k+1,k∈Zx+H=3k+1+{0,±3,±6,±9,.......}x+H=1+{0,±3,±6,±9,.......}x+H=1+H

Let

x=3k+2,k∈Zx+H=3k+2+{0,±3,±6,±9,.......}x+H=2+{0,±3,±6,±9,.......}x+H=2+H

The left cosets of H in Z are H,1+H,2+H .

Answer 14

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

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Answer to Problem 1E

Explanation of Solution

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