Query 22

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course Mth 151

022. `query 22

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Question: `q4.5.9 {-1,0,1} group on multiplication?

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Your solution:

Closed set, identity of 1, 0 has no inverse. Since 0 has no inverse, this cannot be a group.

confidence rating #$&*: 3

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Given Solution:

`a** There are four criteria for the group: closure, identity, inverse property, and associativity.

The lack of any one of these properties means that the set and operation do not form a group.

The set is closed on multiplication.

The identity is the element that when multiplied by other elements does not change them. The identity for this operation is 1, since 1 * -1 = -1, 1 * 0 = 0 and 1 * 1 = 1.

Inverses are pairs of elements that give you 1 when you multiply them. For example -1 * -1 = 1 so -1 is its own inverse. 1 * 1 = 1 so 1 is also its own inverse. However, 0 does not have an inverse because there is nothing you can multiply by 0 to get 1.

Since there is an element without an inverse this is not a group. **

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Self-critique (if necessary): ok

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Self-critique Rating: 3

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Question: `q4.5.25 verify (NT)R = N(TR)

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Your solution:

(NT)R= V R = M

N(TR)= N P = M

confidence rating #$&*: 3

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Given Solution:

`a** From the table

(NT)R= V R = M

and

N(TR)= N P = M

This verifies the identity. **

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Self-critique (if necessary): ok

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Self-critique Rating: 3

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Question: `qquery 4.5.33 inverse of T

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Your solution:

T is its own inverse.

confidence rating #$&*: 3

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Given Solution:

`a** T is its own inverse because T T gives you the identity **

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Self-critique (if necessary): ok

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Self-critique Rating: 3

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Question: `q4.5.42. Explain what property is gained when the system of integers is extended to the system of rational numbers.

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Your solution:

Identity

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`a** The set of integers is a group on addition, with identity 0 and every number x having additive inverse -x.

It is not a group on multiplication. It contains the identity 1 but does not contain inverses, except for 1 itself. This is because, for example, there is no integer you can multiply by 2 to get the identity 1.

If we extend the integers to the rational numbers we do get the inverses. The inverse of 2 is 1/2 since x * 1/2 = 1, the identity. In general the multiplicative inverse of x is 1 / x.

However we still don't have a group on multiplication since 0 still doesn't have an inverse, 1 / 0 being undefined on the real numbers. **

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Self-critique (if necessary):

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Self-critique rating:

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Question: `q4.5.42. Explain what property is gained when the system of integers is extended to the system of rational numbers.

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Your solution:

Identity

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`a** The set of integers is a group on addition, with identity 0 and every number x having additive inverse -x.

It is not a group on multiplication. It contains the identity 1 but does not contain inverses, except for 1 itself. This is because, for example, there is no integer you can multiply by 2 to get the identity 1.

If we extend the integers to the rational numbers we do get the inverses. The inverse of 2 is 1/2 since x * 1/2 = 1, the identity. In general the multiplicative inverse of x is 1 / x.

However we still don't have a group on multiplication since 0 still doesn't have an inverse, 1 / 0 being undefined on the real numbers. **

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Self-critique (if necessary):

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Self-critique rating:

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