Objects with a binary operator

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1.
A group is a set of objects with a binary operator which has satisfy various conditions. Suppose G is a set of elements with a binary operation ⋆. Select each condition which is required for ( G , ⋆ ) to be an group:

For any elements f, g, and h in G, (f * g) * h = f * (g * h).

There exist an identity element e in G such that for all g in G, g * e = e * g = g.

For all elements g in G, there exists an element g-1 in G such that g * g-1 = g-1 * g = e.

For any elements f, g in G, f * g = g * f.

For any elements f, g in G, the element f * g is also in G.

2.
What is the order of the following cycle given in cycle notation (Please enter an integer):

(1 2 3 4 5) (6 7 8)

3.
True or False: The following cycle has an even parity:

(1 2 3 4)

4.
True or False: The cube always has an even parity (the number of cubies exchanged from the starting position is always even).

5.
Find the order of the subgroup of the Rubik’s cube generated by the following element:

(R’ L F R L’ U’)

6.
Which of the following choices is the inverse of the following moves:

R U L R’
Group of answer choices

R’ U’ L’ R

R U L R

R L’ U’ R’

R’ L’ U’ R’

7.

True or False: The following two group elements are the same:
1. U D U
2. U2 D
3. Which of these choices is a valid cycle decomposition of the following permutation?
8
What is the corner/edge parity of the cube after applying the following move to the solved cube? R

Corners are even, edges are even.

Corners are odd, edges are odd.

Corners are even, edges are odd.

Corners are odd, edges are even.

 

Sample Solution

e could be a professional athlete, one who is using their athletic talents to provide for them self and their family. For that person, the injury takes a completely different toll than it does on the high school athlete. It ends their career. It doesn’t just end their athletic career, but their income-earning career. They lose their job and their ability to earn for them self and whomever else they provide for. This, it could be argued, is why there needs to be a way to objectively put these two scenarios into categories by drawing a line.

In cases such as these, the line could be drawn when the pain or result of the pain, in this case the career ending injury, begins to negatively impact day to day functions, as they are synonymous with well-being. If a person cannot perform the simplest of tasks necessary to function throughout their day, their well-being will obviously diminish. It would, therefore, have a negative impact on well-being. Negative impacts on day to day functions can be defined subjectively, because each person’s daily routine varies. Therefore, in the case above, the high school athlete would not have diminished well-being, but the professional athlete would because their everyday functions and routines would change since they would no longer be able to go to work and provide for their family. Other situations this would be applicable in could be two victims of car accidents of differing severity that walk away with different injuries. It could also be applied to people who suffer from differing degrees of depression that impact their daily functions differently. Day to day functions exist to every person, so in any situation that results in pain, drawing the line by evaluating the impact on those functions can subsequently decide if that pain resulted in negative well-being.

In analyzing Colin Klein’s presentation of pain and suffering, he consistently reiterates that pain and suffering are not the same thing. He presents this idea through a series of simple argum

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