Pointed the class at homework 1: [http://www.cs.umb.edu/~eb/370/hw1/] (should have happened last time).
Talked about $\TeX$.
Worked through Euclid's Propositions 1 and 2.
Investigated the proofs behind these propositions, using only previously given common notions, definitions, and/or postulates.
Good (but distracting) discussion of the logical gap - none of the postulates specifies that the two circles in the proof of Proposition 1 must intersect.
Recall that for this proposition, the proof was centered around drawing circles, creating a point at their intersection. This touches on the discussion of the logical gap as mentioned above.
Proposition: On a given finite straight line to construct an equilateral triangle.
Suppose we have two points, A and B (Defined per Def. 1).
Let AB be a line connecting point A to point B (Def. 2).
Describe a circle with center A and radius AB (Def. 15).
Describe a circle with center B and radius AB (Def. 15).
Assuming that, due to the fact that they both have a radius determined by a finite line connection A and B, the circles intersect, describe another point, C, as the point of intersection of the circles (Def. 15).
Let AC be a line connecting point A to point C (Def. 2).
Let BC be a line connecting point B to point C (Def. 2).
Since C lies on the circumference of both circles, and all radii of circles are equal, we can conclude that AC=AB (Def. 15).
Following from the logic above, it must also be true that BC=AB (Def. 15).
Thus since AC=AB and BC=AB, we may conclude that AC=BC (C.N 1).
Therefore, all three lines are equal to one another, and by the definition of an equilateral triangle, the points A,B,C, and the lines connecting the points as described, form an equilateral triangle (Def. 20).
I have taken a class at the university all about proving things. The way that we were proving the propositions were similar in a way. The way that we would use definitions in the reasoning is the way we would prove math in math 280. I like proving things in this class better because i am more of a visual learner so the geometry to Euclid's Propositions is very helpful. We would also use things that we already proved to prove something else. So far i am interested in this class material and the way the class time is being used.
I have to say I did not really take any notes when we were having this class last Thursday, but instead I was more focused on how to think in the 'old' way, and try to process that in my own head, got confused at the beginning. For the propositions we have talked in the class, they were kind easy; the proofs of those propositions are easy to follow and each proposition is kind like adding a bit more onto the previous proposition, I am not sure if that is true, but at least that is how I feel.
I like how we were having the class on last Thursday, unlike math 280, most of us were more 'productive'; it is actually easier to remember what we have learned in class through discussion instead of writing down a bunch of notes. No matter if it is right or wrong, as lone as we have discussion focused on the topic we are study, we know we are thinking.
I did not take too many notes for the second class since all statements of propositions are clear. The propositions need to prove are easy but the way to prove these propositions are not easy at all because we need strong and completely reason to prove these propositions. Like I said in the first class, some propositions are hard to prove just because they state some statement "look" like truth. everyone see the statement will say "oh, that's true" but they don't know why. for example, I know that's a apple but i don't know why it called apple. so it's really hard and important to define the mathematics postulates and propositions at the beginning. once people start with wrong postulates, every proofs following will be wrong.
*Proposition 1: Construct an equilateral triangle on a straight line
*Proposition 2: Construct an equal line from a point. Using Prop 1. Use the straight line connection the given point and the extremity of the given line as the side of the equilateral triangle. Use two circles to give two equal lines which both contains two segments. Use CN 2 to prove the remaining lines are equal.
*Side note: The fact that two circles cut one another is not from Postulates nor CN, but rather from the drawing.