Motivation for Quants. (2) They Turned Your Job Application Down.

One of the most devastating feelings that may hit you, and I mean it!, hit you hard – is the feeling of finding out that your job application has been turned down. Something went wrong and you don’t know what was it.

Bothering questions start flocking to your mind: Have I done everything what it takes to present myself in the best possible way? Was it my fault or they simply were looking for someone else,...

Have you measured your Job conditions and quantify your abilities as in regards to Set Theory? Don't see the point... Well then it's time to start->

1. N itself is countable; just use f(n) = n.

2. N0 = {0, 1, 2, 3, . . . } is countable; use f(n) = n − 1. You can see here why I think that

infinities are funny; the set N0 and the set N - which is its proper subset - have the same size.

3. Z = {. . . , −2, −1, 0, 1, 2, 3, . . . } is countable; now the function f is a bit more complicated;

f(k) = ( 2k + 1, k ≥ 0 −2k, k < 0. ) You could think that Z is more than “twice-as-large” as N, but it is not. It is the same size.

4. It gets even weirder. The set N × N = {(m, n) : m ∈ N, n ∈ N} of all pairs of natural

numbers is also countable. I leave it to you to construct the function f.

5. A similar argument shows that the set Q of all rational numbers (fractions) is also countable.

6. The set [0, 1] of all real numbers between 0 and 1 is not countable; this fact was first proven

by Georg Cantor who used a neat trick called the diagonal argument.

(https://www.ma.utexas.edu/users/gordanz/notes/introduction_to_stochastic_processes.pdf)

And then Read the Sources, Thank you.