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.

8.1-8.2, Due on October 15 (originally due on October 13)

The most difficult part of the reading was understanding that the empty set is a relation because it is a subset of A cross B but it means that not a single element in A is related to B.  Class really did clarify this though.  Other than that it was difficult to understand the notation for domain and range.  I understand what they mean, but the notation looks unfamiliar.

The properties of reflexive, transitive, and symmetry are really important especially in proving things.    If I relate it to chemistry, many reactions are reflexive, only some are symmetric if the equilibrium is 50/50, and practically some are transitive where they can take two steps to reach a final product or only one step, but for the most part they are neither symmetric or transitive.

6.4, Due on 10 October

So 6.4 was difficult to understand.  It seemed very similar to the principle of mathematical induction except there is another step added in where you define where a variable i lives.  It made sense until they made an assumption by the recurrence relation.  In the end, it think it just means showing an added case.

This obviously applies to live because recursion is used a lot by computers and calculators to calculated things numerically. The shorter the recursive formula, the more efficient the calculator, and the quicker that things are processed.  I think that this is the relationship.

6.2, due on 8 October

The most difficult part of the reading was understanding the step with factorials.  It took me a while to remember what a factorial was, and why the book could take the step it did.

The most interesting part of the reading was that it really was not that much different from 6.1.  I wondered why the reading was so short this time.  Most of the examples were proofs that involved the induction theorem and it was pretty neat to be able to reinforce that concept.

6.1, due on 5 October

The most difficult part of the reading was of course all of the algebra.  I have not done common denominators with variables since high school algebra 2.  I look forward to improving my algebra.

The most interesting part was of course the proof by induction.  I remember a Calculus course where after the AP test, we had nothing to do, so our instructor taught us about logical deduction and induction, and told us that induction was almost always better than deduction, but we learn most things through deduction in our every day life.

Midterm 1 Review Due on 3 October

• Which topics and theorems do you think are the most important out of those we have studied?

I feel that the most important topics are the different types of proofs, as well as the proofs that are dealing with real numbers.  It may just be my opinion but determining whether something is even or odd is useful for learning the form of proofs, but I do not see it too applicable in my life.

• What kinds of questions do you expect to see on the exam?

I expect to see a lot of proofs and probably some questions about set operations and truth tables.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
• I still need to work a little bit on the last few chapters about proving something is rational or irrational like for example prove that sqrt(2) is irrational, and also I need to review the set operations in the first few chapters.  Cartesian products are always confusing.  I do not think latex will be on the exam so I should be good there because Latex is simply amazing.