Starting with introducing the idea about One-to-One and Onto in order to define |X| = |Y|. More specifically, Given two sets X and Y, if there exists a well defined function f : X ↦ Y that is 1-1 and
onto, then |X| = |Y|. (Note: the idea of 1-1 and onto is more or less the same as we have learned in Math courses)
Why we want to have this definition? Because it is very helpful for us to understand countability, which is just the "surface" of computability. When counting, we are basically using Natural Numbers. We say the set of Natural Numbers are countable, and then if a set |A| ≤ |N|, then it is countable.
A interesting fact is even if some sets are infinite (e.g. N, or set of odd numbers), they are still countable. So, with the definition, the set of real number should not be countable, because |R| > |N|. For now, I think the idea about computability is still very unclear and abstract, and I expect to learn more about it in future courses.
Honestly, as the last week of the course, my friends and I are not as focus as previously in the class. However, we definitely have learned a lot of new things in this course with the help of professors and TAs. I must thank all of them and also my friends and classmates for the time we spent together in this course, and I wish everyone and myself can get a good score in the final exam!
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