Course Overview

I really like how there are course notes designed specifically for this course but one downfall is the lack of practice questions. I don't expect them to be in course notes or else it would become a full blown textbook. Luckily, there's even a book mentioned in the course notes on page 5.2.

I think what was learned during the course was interesting but I can't contextualize most of the concepts it into my limited understanding of computer science. I came into this class with very little knowledge of computer science. As I come out of this class, I sort of understand the basics of logical notation and the basics of the fundamentals of proof problem solving and format. From what I understand from calculus, this will allow me to read more complex math to understand more math and computer science. The algorithmic analysis has let me peek into the complicated world of the process of writing a solution to a problem before any code is written. As for computability theory, I have no clue how the smart people like Cantor would have thought of using diagonalization to determine there are countable and uncountable infinities and I don't understand it enough to put this info in a context.

This slog ends with a boring bang!

Goodbye!

Although the course notes don't seem that long compared to other classes, the content is more dense.

While I'm reading back, I'm starting to realize just how much can be learned in 12 weeks.


Side note: The course isn't even over yet it's already been useful!

I've noticed this one video that has math logic on a Youtube channel that I watch regularly while procrastinating. The math is at 2:47
https://www.youtube.com/watch?v=Ao0dqJvH4a0#t=167

It's almost over

There's just studying for the exam left and looking back.
The relative shortness of the course notes  doesn't look like a lot in the course notes, I know it took a lot of understanding to get here and I know there's still more to come both in the upcoming week and in the next course.

Lazy caterer's sequence

I searched the problem below online and I found out that it's called the lazy caterer's sequence.

The answer online is  

So I assumed my solution was wrong until I expanded my solution of 1 + (n*(n+1)) / 2

= 1 + (n^2 + n) / 2             #algebra

= (n^2 + n + 2) / 2             #put the one in the fraction

So, YAY!

Unscholarly Sources:

http://en.wikipedia.org/wiki/Lazy_caterer's_sequence

Space Slicing Problem

Problem: If n is some positive whole number, what is the maximum number of regions I can divide a sheet of paper into with n straight lines?

Input: Number of divisions
Output: Max number of regions resulted from divisions.

Plan 1: Draw and find patterns. (I don't know how to show that on this slog)
What I noticed immediately is that each line had to interact with all the other lines.
Also, since the divisions were all lines with different slopes, they could only intersect with all the other lines once.
When the new line went from the boundary of the paper to the first division, it split a region in 2 (or "created" a new area).
However, this thinking didn't really go with the empirical findings. I could only find 10 regions when I drew 4 lines. Then I tried it on the computer and it worked.

Divisions - Regions
1 - 2
2 - 4 (2 + 2)
3 - 7 (2 lines + line going through page -1st line, 1st - 2nd, 2nd and page)
4 - 11 (3 lines: page-1st, 1st-2nd, 2nd-3rd, 3rd-page) or 7 + 4
5 - 11 + 5 = 16

I'm going to quickly guess the answer based on this pattern.
If the number of divisions is n, then the number of regions increases by n every time.
Mathematically, this looks something like: 2 + 2 + 3 + 4 + 5 + ... + n
This looks really similar to 1 + 2 + 3 + 4 + ... + n
and is the same as 1 + (1 + 2 + ... + n)
which is 1 + (n* (n+1)) / 2
and that's my answer for now.

Week 9:

In high school, the week after after a test was usually really chill because there would be very little new material. I guess that's never going to happen until finals. Right after each test, I'm still scrambling to try to catch up all the time. Although I'm learning the stuff in this course, I don't think I'm learning it fast enough.

It's Past Week 8 Mate!

Wow! It's been more than 3/4 of the course already!

I'm getting really nervous about this course with so much new stuff I've never seen before and the future three weeks of even more new stuff that I still have to get trampled by before the exam.

At the end of last week, we got the penny pile problem which I still have no idea how to approach. I am also currently failing to understand the floor function and how to use the definition to prove a modified version of the definition of a limit in Assignment 2.

Week 7: Logic and Math!

Even though it's called mathematical logic according to Wikipedia, it sure doesn't feel like math I'm used to. Set theory with all of it's implications and De Morgan's rules, it feels much more like learning a new language than learning math.

With that being said, I think the math is coming! Prof Heap briefly mentioned how the speed of algorithms are measured with graphs and how they look like the graphs of polynomials. One thing that surprised me is how the speed of algorithms are measured by the worst speed. It doesn't feel like the sturdy math and it reminds me of psychology class where we learned the brain uses heuristics to be faster and sacrifices accuracy.

(I donno) How to Write Proofs

I have a lot of difficulty understanding how these proofs were figured out. I know that most of the proofs in the world just explain how they get from point A to Z but don't explain how they thought of going through all the other 24 points.

In class today, we learned that \for all natural numbers, n² + n is even.

It was done using proof by cases where n² + n had to be even regardless of if n was odd of if n was even.
However, one person pointed out what many were probably thinking in class:
    if n² + n is factorized into n(n+1), this intuitively explains that it must be even because either n or n+1 must be even and any natural number multiplied by an even number is always even.

The problem is that I don't know how to prove this without using the proof by cases and substituting n with 2k+1 or 2k.

End of Week 5: What's the History of Teaching Proofs

I don't think I'm supposed to write down the complete solutions to a problem which leaves me with the question about what to write about. Nothing meaningful usually comes out of my head especially concerning math related topics so I usually look for inspiration while I procrastinate while watching educational videos on Youtube.

https://www.youtube.com/watch?v=S0DSM-EkQE8

Professor Heap has mentioned on numerous occasions on how Euclid established the tradition of proofs.After watching this video, I wonder that if the foundations of proofs that we have started learning in class had been established over 2300 years ago, then are we learning the exact way that every single mathematician has learned proofs? Are we learning the same stuff that Newton did in the 17th century or a royal mathematician lost to history in the Mali empire did?

Cool.

The Folding Problem

The folding problem class was the best class so far in this class for me because I understood what was happening. Obviously, the reason why this class is so challenging for me is because this logic is new to me and I'm not used to thinking so logically. In time, I hope I'll understand.



Also, I apologize in advance

Problem: Folding
Given: # of folds
Required: sequence of up(u) and down(d) creases

Plan 1: (Our only plan) We would go through the folds starting from the first fold. With the new gathered info, we would try to find a pattern and figure out another plan.

Step 1: Count the sequence of ups and downs after each fold

1. d
2. ddu
3. ddudduu
4. ddudduuddduuduu

Step 2: Find the patterns
i) It was discovered that for all n ∈ number of folds, the numb er of creases is 2ⁿ -1

1. d (2¹ -1)
2. d    du (2² -1)
3. d    du    dduu (2³ -1)
4. d    du    dduu    ddduuduu (2⁴ -1)

ii) A group member also discovered that there is always one more down crease than up crease.With these two observations, predictions for the next folds

Folds	Down	Up
1	1	0
2	2	1
3	4	3
4	8	7
5	16 (25 -1) / 2	15 (25 -1) / 2 - 1

iii) Here, we were stuck because the patterns above did not predict the sequence of ups and downs. So, we tried to get more info by labelling which creases were created as a result of which fold number.

It was discovered that every new fold created new creases in a u,d,u,d order.
For example, Fold 2 created u,d
Fold 3 created u,d,u,d  or 2  of each
Fold 4 created u,d,u,d,u,d,u,d, or 4 of each
Combined, it looks like this:
                              1
                              d

              2              1               2
              d              d               u

      3      2      3      1      3      2     3
      d      d      u      d      d      u     d

  4  3  4  2  4  3  4  1  4  3  4  2  4  3  4
  d  d  u  d  d  u  u  d  d  d  u  u  d  u  u

Following this trend of putting the new number in the spaces between all the creases, the next sequence should be:
5453545254535451545354525453545
dduuddudddudduuddduuduuuddudduu

I can't figure out a way to predict the sequence of a large number of folds.

And because I can't draw this out, here's the dragon curve:
https://www.youtube.com/watch?v=NajQEiKFom4

Welcome!

Welcome to my blog! There most likely won't be any content that's more creative than all of the other SLOGS so I wish you the best of luck on your journey to try to read this boring slog!

This is the first time I have ever been exposed to logic formally and it intrigues me how counter intuitive logic can seem at first. Taking the example of the vacuous truth, it seems bizarre that if the universal quantifier describes an empty set, the second statement is always true.

It only makes sense after taking a step back and taking time to let it sink in. This is probably a sign of what's to come.