Section 32 due 26 November

This proof is so much more complicated than I thought it would be.

The doubly infinite chain reminds me of how we are taught that there is no beginning nor end to the Gods. Thus each God came from an earth ruled by a God who came from an earth... etc. There is no beginning. It is hard to understand the implications of that.

"[T]he restriction of an injective function is still injective" is such a nice trait.

How does example 32.3 serve as an example of the Schroder-Bernstein theorem? I don't really see how it makes use of the theorem.

Section 31 due 20 November

Is the best way to disproof the lack of a bijection to assume there is a bijection and seek a contradiction? Or should we generally try to find some element that illustrates that the function is not surjective?

I am confused about the proof of the |S| < |P(S)|. Also I'm not following how the barber analogy relates...

Is N the biggest countable set? (And all sets whose cardinality is equivalent to |N|?)

-Batman

Section 30 due 19 November

My intuition tells me that the proof of theorem 30.4 is incomplete. We can continue the pattern to infinity in theory, but because we never actually get to infinity we have no way of knowing down the line if the resulting decimal expansion is truly not achieved in our function... I am going to have to retrain my intuition.

I do not understand the proof that (0,1] has continuum cardinality. What is the purpose of S?

That was a short section...

-Batman

Section 29 due 16 November

How is the diagram used? Is it merely a heuristic for how to count in a way that gets every possible combination exactly once? If so, then it makes sense, though I'm not sure why this is considered a "clever method." It seems pretty obvious, assuming I understand it correctly.

The proofs addressing the rational numbers are interesting.

-Batman

Section 28 due 14 November

WOAH!!! Crazy proofs are happening. See theorem 28.4. That's pretty epic. Though I wonder how we can conclude that the proofs must have the same size, considering the table only portrayed a subset of each set. How do we know that one set doesn't "run out" of elements to pair with the other set?

I was going to ask you if it is an adequate proof of bijection to simply find an inverse function between the same elements as the function. Then I forgot. Then I remembered, but your office hours were super full. I'm hoping this will remind me to ask you when I see you tomorrow.

I would like to settle a debate I was having with a classmate who I will leave unnamed. Take an infinite set and divide it into two subsets such that each subset is mutually exclusive and that between the two of them you cover every element in the set. Can one of these subsets be finite, or do they both have to be infinite sets?

This is super exciting material. I am eager to start the homework.

Section 27 due 12 November

The implications of 27.3 are cool.

Theorem 27.5 is useful. I did not know that, though it is logical.

Woah, surjective reductions are interesting. Are they related to projections in linear algebra?

It keeps messing me up that the range of the function is called the "image." Grrr...

Preimages are cool.

Section 26 due 9 November

I did not know the properties in theorem 26.12. I'm not sure I follow their proofs.

I find it interesting that there are so many instances of an "identity" object in math. For example, $I$ is the identity matrix that does not change any matrix it is multiplied by. In scalar multiplication, 1 times anything equals anything. In addition, 0 plus anything equals anything. Now we have an identity function. I seems somewhat counterintuitive that something that doesn't affect whatever it is applied to is such a powerful tool in math. That's what makes math interesting!

It would have been very nice to have known the technical definition of the inverse relation at my math camp. It would have made my life easier. Now I'm salty.