Lecture 2: Cantor's Theory of Cardinality (Size)

MIT 18.100A Real Analysis, Fall 2020
Instructor: Dr. Casey Rodriguez

What does it mean for one set to be bigger than another? Defining injections, surjections, bijections, and cardinality, and showing the countability of the rationals.

License: Creative Commons BY-NC-SA

[mahmudurrahman]

Now we need a course on Complex analysis, Partial differential equations and differential geometry

[roberthuber]

I find it really helpful to pause and try to proof the theorems before they are worked out in lecture.

[rajinfootonchuriquen]

It's really crazy that rational, integer and natural sets are all the same cardinality

[egomezpele]

Great lecturer, pedagogically is the best I've seen to explain such concepts. It's important to really understand the basic concepts to lay the foundations. If you take for granted some basic facts probably sooner or later you will pay the price, you really need to understand why it's true. You need the graphic intuition to guide you through the crude Algebraic/Logic demonstrations, stick to the definitions and apply the techniques for demonstrations. And this is probably one of the reasons why United States has the best Universities

[alexanderpopov]

Thanks to MIT ocw for another great course. Hoping to see more courses in fundamental mathematics in the future

[casper]

1:01:09 Finding mappings in a creative way to show that seemingly larger sets have the same cardinality as seemingly smaller sets reminds me of the paradox of the infinite hotel by Jeff Dekofsky. In this paradox, clever mappings are also created to keep finding hotel rooms no matter how many guests arrive.

[KirbaeK]

this video is really rocking my socks off

[gloriosatierra]

Cantor is by far my favorite Mathematician.

[napluto]

Maybe the cameraman can zoom in more, profs handwriting is a bit small

[enisten]

According to the course calendar, Assignment 1 is due on the same day as Lecture 2, i.e. this lecture. But this lecture contains material that is needed to be able to work on Assignment 1. In fact, towards the end of the lecture, the prof gives a sketch of how to solve Exercise 6 of Assignment 1. Why is this?

[eccotom]

lecture 1: 60k views
lecture 2: 6k views
lecture 3: ...

[adamcai]

1:06:27 it shall be q_M^{s_M} instead of q_M^{s_N}

[darkpikachu_.]

Please upload videos on measure theory too🙏😊

[Neemakukreti]

Can one prove the assertion at 58:35 via the absolute value function...instead of the one professor suggested

[jdjdiifk]

at 54:55 why is it necessary to add a dummy variable in proving that the function is surjective. when doing other problems which are more complicated I found that it's a lot more simpler if there are no dummy variables in the proof, but I'm not sure if that makes the proof incorrect if there aren't any dummy variables.

[Javy_Chand]

I could NOT prove the second part of the bijection proof on the Assignment, that is, 6. b), using f(q) = aka the positive non-natural numbers part of the function, as it is a piecewisely-defined function, you need to prove bijectivity on both sides.

Alas, I got stuck in proving it, since I could not get an expression in terms of x_1 and x_2 (or q_1 and q_2)
The Natural part was quite trivial, but I just cannot see how to prove the rational-only part.

My guess is there should be two quadratic terms, we take the square root and be left with absolute values, and since q>0 that implies that only one branch makes sense, but I just cant work out the problem properly.

If anyone is kind enough to share a solution, I would appreciate it!

[georgelu]

It is so stupid that the camera is moving around and always fails to center toward the blackboard.

[Jhopsssss]

Why does he say his definition is ambiguous? In what ways does defining a function as some subset of AxB make the definition less ambiguous?

[cswanson]

1:16:32 The answer is YES, of course. But how?

[funnywarnerbox]

I could not prove the rationals are countable by proving the function at 1:08:30 is a bijection. Has anyone done this?