Abstract Algebra: A NEW Start – Baiting

Retrieved from https://www.booktopia.com.au/abstract-algebra-gerhard-rosenberger/prod9783110250084.html

In this blog, I would like to briefly review my Fall semester and share my plan for the Spring.

In the past four months, I worked on Differential Equations and its related topics. Through following the MIT Open Course Ware, I learned different methods to solve DEs and their implications in real life.

In the Spring, I will move on from DE to a collection of new topics. In specific, I will spend some time on Modeling the Future (MTF), a mathematical modeling competition that focuses on the medical and insurance industry. Since MTF is a team competition, I will communicate with my teammates and work on the data, modeling, and written report together. In my future blog posts, I will share more updates about this competition, but it will not be the focus of my blogs.

I have also started preparing the Chester County Science Fair. This year, I am aiming to present a theoretical-based project. Currently, I am working on Abstract Algebra. Since Abstract Algebra is a college-level course, I am not planning to strictly follow any textbook. Instead, I will use textbooks as references to learn some basic concepts and move on to explore the knowledge that attracts most of my interests.

Aside from these two topics, I will try to find time and self study some Computer Science. Due to my rather poor foundation on coding, I will follow an open course from [edx.org]. As I currently want to devote my full efforts to the two upcoming competitions, I will postpone my learning on CS for about a month.

Abstract Algebra

I first encountered Abstract Algebra through the following joke:

[A guy asks a French primary school student: “Do you know the answer of 2+3?”

The student replies: “No, I don’t.”

“Then what did you learn in your classes?” The guy asks, slightly surprised.

“I learned 2+3=3+2.”

“Why?”

“Because real numbers form an Abelian Group in addition!” The student answers.]

Well maybe I no longer remember the joke so well, but you get the point. And if you have heard of Nicolas Bourbaki Group and their education reformation, you will probably evoke laughter. What they tried to do was to teach the “real math” to students since the very beginning. In other words, if this reformation happens at Westtown, we will all be taking Calculus first and then take Algebra and Geometry.

Bourbaki congress1938.png
Photo was taken in the Bourbaki Congress of 1938.

If this sounds a little confusing, here is an example:

We all know 2+3=5 and 3+2=5, so 2+3=3+2. This is called elementary algebra. In the same logic (well it’s not literally the same logic, but I will use this term for now), 2*3=3*2.

But why?

Because of commutativity and distributivity, you may say.

But why?

Now you see, our conventional math fails. To answer this question, mathematicians had to redefine addition.

And they created the concept of group, binary calculation, loop and thousands of things that make us a headache.

And they tried to teach these stuff to primary school students.

Well, Bourbaki’s idea may sound interesting or stupid to you, but what I learned from the joke was the attitude to keep asking and answering “why.” So I started to work on Abstract Algebra.

Since I am not learning the course structurally, I first started to look at mapping, a concept also taught in Linear Algebra.

If A and B are two non-empty set, and there exists a transformation f such that each A has one and only one corresponding B through f, then we call this the mapping from A to B.

Notice that even though each A has exactly one corresponding B, each B might have multiple or no corresponding A.

For instance, if A={A0, A1, A2, A3, A4}, B={B0, B1} and

  • f(A0) = B0
  • f(A1) = B0
  • f(A2) = B0
  • f(A3) = B0
  • f(A4) = B0

It is still a mapping.

However, according to the definition, if one of the following happens, then this is no longer a mapping:

  1. f(A0) = B0 or B1 (an A has more than one corresponding B)
  2.  f(A1) = k, and k ≠ B0, k ≠ B1 (an A has no corresponding B)

This leads to two special cases:

If f(A) = B, then we call this mapping a surjection.

If f(Ai) ≠ f(Aj) when assuming i ≠ j and Ai ≠ Aj, then we call this mapping an injection.

And if the mapping is both a surjection and an injection, then we can simply call it a bijection. If you want to visualize these relationships, please refer to the this graph.

Related image
http://www.bartuavci.com/2014/07/bijection-injection-and-surjection.html

For a bijection consists of set A and B and f(A) = B, there must also exist a transformation g (g may equal to f) such that g(B)=A. So obviously,  gºf(A) = A and fºg(B)=B.

This is where I left off for the past week. As I said, I am incredibly passionate about Abstract Algebra as it further answers the questions can’t be solved using conventional algebra. In the next week, I will talk about closure, identity element, and inverse element. Furthermore, I will share my progress on both of my competitions. In my future blogs, I will also focus on more math and their applications in real life.

Works Cited

“Abstract Algebra.” booktopia, 28 Jan. 2011, www.booktopia.com.au/abstract-algebra-gerhard-rosenberger/prod9783110250084.html. Accessed 27 Jan. 2019.

Bijection, injection and surjection. 11 July 2014. O. Bartu AVCI, www.bartuavci.com/2014/07/bijection-injection-and-surjection.html. Accessed 27 Jan. 2019.

Bourbaki congress at Dieulefit on 1938. Wikepedia, 26 Jan. 2019, en.wikipedia.org/wiki/Nicolas_Bourbaki#/media/File:Bourbaki_congress1938.png. Accessed 27 Jan. 2019.

“代数结构入门:群、环、域、向量空间” [“An Introduction to the Algebra Structure: Group, Loop, Field, and Vector Space”]. Spark & Shine, 24 Feb. 2015, sparkandshine.net/algebraic-structure-primer-group-ring-field-vector-space/. Accessed 27 Jan. 2019.

4 thoughts on “Abstract Algebra: A NEW Start – Baiting

  1. Dhillon

    Hi Baiting, I am really excited to see where this new semester will take you! Your questions regarding simple addition and the funcitonality of it is very fascinating to me. I think this will be a great topic to bring up in our Religion and Science class!

    Reply
  2. anindywp

    Hi Baiting, Abstract Algebra seems quite interesting! I wonder if mathematical logic would impact on higher level computation on coding algorithms (best solution). I can tell that it makes simple math harder than they look: math is a truly fascinating subject!

    Reply
  3. sabrina.schoenborn

    Hey, Baiting! This seems like an incredibly interesting topic! How/Do you think you’ll share this with the community in any way?

    Reply
    1. baitingz Post author

      I am also working on this topic for my science fair. I am planning to redefine some of the concepts (like what is addition, multiplication, etc.) in math and using my definition to prove the commutativity and closure. Then (if I can figure out how to do this), I will try to prove Lagrange’s Theorem, and eventually Fermat’s Little Theorem. This is a long way, but hopefully I can get it done…

      Reply

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