If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

022.  `query 22

Question:  `q4.6.9 {-1,0,1} group on multiplication?

Your solution: 

Confidence Assessment:

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.  **

Self-critique (if necessary):

Self-critique Rating:

Question:  `q4.6.25  verify (NT)R = N(TR)

Your solution: 

Confidence Assessment:

Given Solution: 

`a** From the table

(NT)R= V R = M

and

N(TR)= N  P = M

This verifies the identity. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qquery 4.6.33  inverse of T

Your solution: 

Confidence Assessment:

Given Solution: 

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

Self-critique (if necessary):

Self-critique Rating:

Question:  `q4.6.42.  Explain what property is gained when the system of integers is extended to the system of rational numbers.

Your solution: 

Confidence Assessment:

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. **