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course mth 151

Assignment 4.5

022. Groups

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Question: `q001. There are nine questions in this assignment.

A group is a set and an operation on that set which has the properties of closure, associativity, identity and inverse. The set {1, 2} on the operation @ of Assignment 21, defined by x @ y = remainder when the product x * y is divided by 3, does have the associative property. Is the set {1, 2} a group on @?

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Your solution: Yes

confidence rating #$&*: very confident

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Question: `q002. Which of the properties closure, identity, commutativity, inverse, does the standard addition operation + have on the set {-1, 0, 1}?

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Your solution: identity, commutativity and inverse

confidence rating #$&*: very sure

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Question: `q003. Does the operation * of standard multiplication on the set {-1, 0, 1} have the properties of closure, identity and inverse?

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Your solution: it does have identity and closure

confidence rating #$&*: fairly confident

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Question: `q004. Does the operation * of standard multiplication on the set {-1, 1} have the properties of closure, identity and inverse?

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Your solution: yes, it has all three - closure, inverse and identity

confidence rating #$&*: very confident

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Question: `q005. Is the operation * of standard multiplication on the set {-1, 1} a group. Note that the operation does have the property of associativity.

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Your solution: yes, it is a group

confidence rating #$&*: very confident

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Question: `q006. We've referred to the property of associativity, but we haven't yet defined it. Associativity essentially means that when an operation (technically a binary operation, but don't worry about that a terminology at this point) is performed on three elements of a set, for example a + b + c, it doesn't matter whether we first perform a + b then add c, calculating (a + b) + c, or group the b and c so we calculate a + (b + c). If + means addition on real numbers, show that (3 + 4) + 5 = 3 + ( 4 + 5).

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Your solution:

(3 + 4) + 5 = 7+5 = 12

3+ ( 4 + 5) = 3 + 9 = 12

This has associativity

confidence rating #$&*: very confident

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Question: `q007. Verify that for the operation @ defined on {0, 1, 2} by x @ y = remainder when x * y is double then divided by 3, we have 2 @ (0 @ 1) = ( 2 @ 0 ) @ 1.Verify also that (2 @ 1) @ 1 = 2 @ ( 1 @ 1).

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Your solution:

A)0@1=0 ==> 2@(0@1) = 2@0 or 0

2@0=0 ==> (2@0)@1 = 0@1 or 0

THerefore

2@(0@1) = (2@0)@1

B)(2@1)=1 ==>(2@1)@1=1@ or 1

1@1=2==>2@(1@1)=2@2=2

Therefore

(2@1)@1 = 2@(1@1)

confidence rating #$&*: confident

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Question: `q008. Does the result of the preceding exercise prove that the @ operation is associative on the set {0, 1, 2}?

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Your solution: Yes

confidence rating #$&*: very condifent

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Question: `q009. Earlier we verified the properties of closure, identity and inverse for the multiplication operation * on the set {-1, 1}. We asserted that this operation was associative, so that this set with this operation forms a group.It would still be too time-consuming to prove that * is associative on {-1, 1}, but list the possible combinations of a, b, c from the set and verify associativity for any three of them.

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Your solution:

(-1, -1, -1) (-1, -1, 1) (-1, 1, -1)

A(-1 * -1) * -1 = -1

B(-1 * -1) * 1 = -1

C (-1 * 1) * -1 = -1

confidence rating #$&*: very confident

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022. `query 22

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

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Your solution: not a group on multiplicaition

confidence rating #$&*: very confident

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

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Your solution:

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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):

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Question: `qquery 4.5.33 inverse of T

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Your solution: T is it's own inverse

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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: inverse

confidence rating #$&*: not confident"

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Self-critique (if necessary):

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Question: `qquery 4.5.33 inverse of T

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Your solution: T is it's own inverse

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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: inverse

confidence rating #$&*: not confident"

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Self-critique (if necessary):

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Self-critique rating:

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