Pre-Test post, due on October 31

Which topics and theorems do you think are the most important out of those we have studied?
I think it's most important to remember the regular and strong principles of mathematical induction, because they are from the very beginning of the unit and we may have forgotten the specifics. It's also very important to remember what makes relations equivalence relations or functions. Finally, being able to prove something is injective or surjective is very important.

What kinds of questions do you expect to see on the exam?
I expect questions asking me to prove statements through induction, as well as questions that ask whether some relation is reflexive, symmetric, or transitive. Also, there will be questions about proving statements that deal with the integers mod n and their equivalence classes. We will need to show when a relation is and isn't a function, and whether it is injective or surjective. There will be questions about composite and inverse functions, and maybe a few dealing with permutations.

What do you need to work on understanding better before the exam?
I need to review inductive proofs, and the definitions for relations, equivalence relations, functions, equivalence classes, etc. I need to remember what it means for a set to be well-ordered.

9.6-9.7, due on October 29

What was the most difficult part of the material for you?
I didn't at first understand what the section was talking about when it described how we can create a new function that has the same rule but different codomain, so that it then has an inverse. The example of e^x and ln(x) really helped me understand this.

What was the most interesting part of the material for you?
I liked learning about permutations, because I already learned about them on a surface level in high school and it's interesting to look at them from a more complex mathematical perspective. Permutations here apply in a lot more places than permutations as I learned in high school.

9.5, due on October 27

What was the most difficult part of the material for you?
I had a hard time understanding how composition of three functions could be associative. I looked at an example and it made a lot more sense because I realized it doesn't matter which function you "plug in" to the other function first, because the order is the same.

What was the most interesting part of the material for you?
It's interesting to think about when operations on functions make sense and when they don't. For instance, multiplication and addition make good sense on the set of all real numbers, but when we have an arbitrary set of elements we couldn't say that one element multiplied by another is an element of the set.

9.3-9.4, due on October 24

What was the most interesting part of the material for you?
I liked learning about one-to-one functions because from calculus I remember that the inverse of a one-to-one function is still a one-to-one function (when the functions are on the real numbers). It's cool that we can prove functions to be one-to-one.

What was the most difficult part of the material for you?
I didn't get the definition for surjective functions at first because I couldn't remember the relationship between codomain and image. I think an easier definition for a surjective function is when the codomain is equal to the range.

9.1-9.2, due on October 22

What was the most difficult part of the material for you?
For a minute I didn't understand the difference between a codomain and a range. A codomain is the set of objects that could be in the range (like all real numbers for f(x)=x^2).

What was the most interesting part of the material for you?
It's cool to think about the set of all functions from one set to another. Usually when we talk about functions we talk about functions with infinite domains. But when the domain and codomain are finite, it actually does make sense to talk about this, and talk about the length of it as well.

8.6, due on October 20

What was the most interesting part of the material for you?

Understanding when an operation is well-defined seems like an interesting topic, because it's interesting how the definitions of addition and multiplication we've used are well-defined. It seems intuitive at first, and the proofs help it to make more sense.

What was the most difficult part of the material for you?

I don't understand why alternative operation definitions would be useful, especially when they aren't well-defined. The example at the end of the chapter seems random and not useful.

8.5, due on October 17

What was the most difficult part of the material for you?
At the very end of the section I had a hard time understanding why not all equivalence relations in terms of the integers mod n have n distinct equivalence classes. The example helped that fact make a little more sense in my mind. Maybe all linear congruence mod n relations have n distinct equivalence classes?

What was the most interesting part of the material for you?
I think it's interesting to study a new definition of congruence modulo n that uses our understanding of relations. I think it brings a deeper understanding of congruence than we had before.

8.3-8.4, due on October 15

What was the most interesting part of the material for you?
Theorem 8.3 was interesting to me because that the equivalence classes of R make up a partition of the set R is defined over. It's interesting that equivalence relations have that property.

What was the most difficult part of the material for you?
I had a difficult time understanding why equivalence relations exist that don't exactly have an equation that describes them. I suppose using the definition that we have is the most useful.

8.1-2, due on October 13

What was the most interesting part of the material for you?
This section is cool because relations seem to be a very abstract idea of a function in 2- or 3-space that we deal with in calculus. Now we consider a more general idea; it's interesting to think about how two sets are related at the most basic level.

What was the most difficult part of the material for you?
Understanding what made a relation transitive was difficult in the same way that understanding the truth table of an implication was difficult. For a relation to be not transitive, we have to show an example where (x,y) and (y,z) are elements of R, but not (x,z). So it's actually a stronger statement to say a relation is not transitive than it is to say that it's transitive.

6.4, due on October 10

What was the most interesting part of the material for you?
I thought the proof at the end of the section was interesting, because I didn't quite see how the writer could see where to go with the proof. At the end, it suddenly made sense, and now I understand how to think about these kinds of proofs.

What was the most difficult part of the material for you?
At first I didn't see why we would ever need to use the Strong Principle of Mathematical Induction. But until now I've only seen exercises where it was simple enough to get from the assumption to the conclusion of the induction step. I didn't realize there would be proofs that would require more assumptions to reach the conclusion.

6.2, due on October 8

What was the most difficult part of the material for you?
Sometimes it's difficult to understand how to show that if P(k) is true then P(k+1) is true. There's a certain methodology to knowing how to start these problems that will take practice.

What was the most interesting part of the material for you?
I think it's cool that we can prove things in math both with deduction and induction. I've always thought of experimental science as the only inductive science, and theoretical science and math as only deductive. This is obviously not true.

6.1, due on October 6

What was the most interesting part of the material for you?
I think it's interesting that the set of natural numbers cannot be proven to be well-ordered. I wonder if there is a proof showing that a proof of this is impossible.

What was the most difficult part of the material for you?
These proofs by induction generally contain much more algebra than other types of proofs, so it's easy to get lost in the algebra. I need to make sure I follow each step carefully and correctly as I prove things by induction.

Questions, due on October 3

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

The most important topics include the methods of proofs (direct, contradiction, contrapositive). The basic definitions of sets are also important, as are truth tables and understanding when statements are equivalent.

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

I expect some questions about set operations, truth tables, statements, biconditionals, and proofs by various methods.

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 need to be sure to remember theorems like De Morgan's laws, so I can convert statements to simpler form in proofs.

My math question:

Prove that π is irrational.