CONTEMPORARY MATHEMATICS TEST II
THURSDAY 11Ah MARCH 2004 : TAKE HOME TEST DUE : TUESDAY 23rd MARCH
Question 1
Use the diagram below to complete Table I as a binary operation of the transformations
indicated below on the equilateral triangle ABC for set T = {X, Y, Z, R, S, I}
A
y
z
B
C
x
Let the center of the triangle ABC be O.
X means ‘Reflect in line x’
Y means Reflect in line y’
Z means “Reflect in line x’
R means ‘Rotate by 120 degrees anticlockwise about O’
S means ‘Rotate by 240 degrees anticlockwise about O’
I means ‘Do nothing’
i)
X
Y
Z
R
S
I
X
Y
Z
R
S
I
1
Table I
ii)
Is the operation (transformation) closed?
iii)
Give reason(s):
iv)
Is the transformation associative?
v)
Give at least one example to demonstrate the associative property.
vi)
Is one example sufficient to establish the associative property?
Give reasons.
vii)
Is the commutative property valid for all members of set T?
viii)
If no, give a counter-example to illustrate your reason.
ix)
Is set T a group?
x)
State the four conditions of a mathematical system that is a group.
State how the set T meets the four conditions.
xi)
State the identity member of set T.
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xii)
If the identity exists, state the inverse element of very member of set T.
Element
X
Y
Z
R
S
I
Inverse
Table II
Question 2
Use the article to answer question #2.
Reference: M. Jones & S. Cohen. “Fairness: How to Achieve it and How to Optimize in a
Fair-Division Procedure”. Mathematics Teacher 97, (March 2004): 170-175
Use the FOUR conditions of envy-free fair-division procedures discussed in class to
divide the items listed in Fig 1 of the above article between Ann and Bob.
Consider the bidding numbers as the percentage values attached to the items by Bob and
Ann.
Please give a good presentation. Brief and concise solutions are better than long and
meaningless answers. Justify how your answers satisfy the criteria established for envyfree fair division. A type written solution is required for question #2.
Do you have the same result as the authors? Mention the similarities and dissimilarities
between your solution and the authors.
Do you think your division is fair?
Present a convincing argument to establish how fair your decision is.
Can you suggest another fair division process to help you achieve better results?
You are expected to discuss your solution in class as part of your assessment. Be
prepared. Thank you.
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