Transitive Properties

Mathematics 100: Geometry and Logic
Final Examination


Question: What happens when you draw lines through a pre-existing entity?

Answer: If you take an object (say, an unbroken circle) and bisect it at its widest point with a straight line, you will create a new object made up of congruent halves. For our purposes, we will call these compartments. The space once taken up by the object will also be lessened, due to the presence of the straight line. Note also that the extent of this diminishment will depend on the amount of space alloted to the straight line. If you bisect this straight line with another straight line, at a right angle, you will create a cross. This cross will diminish the size of the original compartments, and through careful analysis, you will be able to prove that the compartments can now only relate to one another in terms defined by the cross. Please also observe that while the resulting compartments have similar characteristics, they cannot join until the cross is overcome and the straight lines have been erased.

- End of Proof.

Before beginning the following question, please take time to review the following definitions:

Transitive Property of Equality:

"If a = b and b = c, then a = c.

The Transitive Property is one of the equivalence properties of equality. This is a property of equality and inequalities. One must be cautious, however, when attempting to develop arguments using the transitive property in other settings."

Lemma

"A helping theorem. A lemma is proven true, just like a theorem, but is not interesting or important enough to be a theorem. It is of interest only because it is a stepping stone towards the proof of a theorem."

Postulate:

"A statement accepted as true without proof. "

Axiom

"A statement accepted as true without proof. An axiom should be so simple and direct that it is unquestionably true. "



Question: Ben likes boys. What does that make Ben?

Answer

1) Ben likes boys. Therefore, Ben is gay (see lemma “if you are a boy and you like another boy or boys you are gay” ).

2) Gay is bad ( one can infer this from the answer to the preceding question, or by using the popular "gay is bad" postulate).

3) Therefore, Ben is bad. (Transitive Property).



Bonus question – if you get this one right, you will pass the whole course, regardless of the work you have done before…

Ben is bad. Now what?


We have proven that Ben is a bad object. However objects have many definable properties and we can observe that Ben is also good in school and can play the piano and sing. He can also cook and imitate his parents in a way that makes people laugh. If Ben works at these things hard enough for a long enough period of time, it is logical that people will overlook his inherent evil characteristics.

However, there is a new concept which is really a very old concept which supposes that all objects are good. While this concept cannot be proven, per se, we find that if adopted, it becomes self-evident. It is therefore an axiom, though it is viewed by some as experimental, controversial and vulnerable. We believe it to be correct.


Therefore,

Ben,

Who likes boys,

Is good.

He will still play the piano and sing and make fun of his parents, but he will not do it so that others will overlook any other aspect of his being. In this way, we can assume that Ben, in all probabilty, will be observed in a more comprehensive way, and can therefore exist in a more cohesive manner.


- End of proof.