#$&* course Mth 151 023. `query 23
.............................................
Given Solution: `a** There are four criteria for the group: closure, identity, inverse property, and associativity. The lack of any one of these properties means that the set and operation do not form a group. The set is closed on multiplication. The identity is the element that when multiplied by other elements does not change them. The identity for this operation is 1, since 1 * -1 = -1, 1 * 0 = 0 and 1 * 1 = 1. Inverses are pairs of elements that give you 1 when you multiply them. For example -1 * -1 = 1 so -1 is its own inverse. 1 * 1 = 1 so 1 is also its own inverse. However, 0 does not have an inverse because there is nothing you can multiply by 0 to get 1. Since there is an element without an inverse this is not a group. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `q4.6.25 verify (NT)R = N(TR) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you look at the table on p.213 of the book, (NT) = V * R = M N(TR) = N * P = M They both equal M confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): OK ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qquery 4.6.33 inverse of T YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** T is its own inverse because T T gives you the identity ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK, though I didn’t explain ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q4.6.42. Explain what property is gained when the system of integers is extended to the system of rational numbers. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: You gain the ability for inverses. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** " 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: You gain the ability for inverses. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The set of integers is a group on addition, with identity 0 and every number x having additive inverse -x. It is not a group on multiplication. It contains the identity 1 but does not contain inverses, except for 1 itself. This is because, for example, there is no integer you can multiply by 2 to get the identity 1. If we extend the integers to the rational numbers we do get the inverses. The inverse of 2 is 1/2 since x * 1/2 = 1, the identity. In general the multiplicative inverse of x is 1 / x. However we still don't have a group on multiplication since 0 still doesn't have an inverse, 1 / 0 being undefined on the real numbers. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&! ********************************************* Question: `q4.6.42. Explain what property is gained when the system of integers is extended to the system of rational numbers. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: You gain the ability for inverses. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!#*&!