9.6-9.7, Due on 29 October

The most difficult material in the reading was that on permutations.  I understand the concept that the permutations are simply reordering the numbers, but it still maps to the original domain.  Once the book started using matrices though it reminded me of linear algebra.

The most interesting part was how the book worded inverses.  I remember seeing these functions but we would always set them up as matrices in order to find the inverse.  I do not remember exactly how we found the inverse that way.  I do remember in high school just switching y and x and that seemed to always work.

9.5, Due on October 27

The most difficult part of the reading was the proof on a composition being injective and subjective.  I have been doing so much studies in other courses, that I barley remembered the meaning of those two terms as they are pretty new to me, but the proof did not seem to be that difficult.

The most interesting part was all of the applications in calculus including the product rule and the chain rule.  I thought it was cool how they were mentioned and was hoping to see more, but this book probably does not cover calculus.

9.3-9.4, Due on 23 October

The most difficult part of the material was understanding why the cardinality of A had to be bigger than the cardinality of B for a function to be one to one from A to B.  After re-reading the section it became obvious that if B was less than A, then there would have to be repeating values and it would no longer be one to one.

The most interesting part of the material was simply the identity function expressed that way.  I am so used to seeing it as x=x or something similar.  I cannot wait to use this function in proofs, but it seems almost trivial.

9.1-9.2, Due on 21 October

The most difficult part of the reading was understanding what an image is.  I eventually realized it is just the value that you get when you plug something into a function.  The way the book stated it was not the most clear to me.  Example 9.1 was also a little difficult to understand.  I thought they were talking about related functions, when they were just giving a list of examples.

The most interesting thing was mapping.  I remember hearing mapping a lot especially when doing triple integrals.  We would either map a function to a plane, or map it to polar coordinates so that we could do the integral.  It wasn't too hard once we got some practice with it.

8.6, Due on October 19

The most difficult part of the material, was the principle of what makes something well defined and the proof of n greater than or equal to 2.  I think I understand the principle that the equivalence classes need to be equal, but I am not very confident in my ability to prove it.

The most interesting part of the material was the principle of closed to addition and multiplication.  I remember seeing those principles before, but I am happy now to have a way to define them using these new equivalence classes.

8.5, Due on 17 October

The reading was not all that long and pretty straight forward.  The most difficult part was understanding the different equivalence classes for modules and following their work on how to get there.  The classes make after looking at it a while though.

The neatest part of the material was that modulos are reflexive, transitive, and symmetric, which is interesting.  At just a glance it would not seem so.  I still do not understand why they are useful though to be honest.  The book said something about a division algorithm that we will study later.  I suppose I will just have to wait.  I just know how to prove that they are equivalent.

8.3-8.4, Due on October 15

The most difficult part of the reading was the section over equivalence classes.  The proofs for transitive, symmetric, and reflexive really are not all that new, but the analysis of equivalence classes took a while to understand.  I am interested in what they are useful for as I understand they are partitions.  I assume we will learn later.

The most interesting part of the material that I got to read was simply the new definition of equivalence as something that is reflexive, transitive, and also symmetric.  I had never though of equivalence in that way, but it makes sense.  I also thought it was interesting they used the word class in equivalence class.  I do not recall anything being called a class in math, and now that I think about it, it is not often used in physics or chemistry.  In biology of course we have kingdom, phylum, class, genus, order, family, species, but that is not a very mathematical definition but more of an arbitrary classification.