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Hi, so I'm enjoying the book so far and a lot of it is going in at the minute. I'm finding writing examples down and then testing them in code to be an effective way at gaining intuition.
However I'm struggling with correctly creating real-world examples in exercise 2.2.3:
I originally tried a 3rd degree polynomial i.e.
but thought maybe I should've tried something easier so I went with a 2nd degree.
My assumption that if and
where and , therefore

So for a real-world example:

If concentrate on the b coefficients then and

I believe then just become 8

This is where my abilities and conception of maths start to disrail:
I try to get

So then it's
Which then starts to hurt the head...
At this point I decide I've gone down the wrong path and I've made a (probably obvious) mistake somewhere, I'd appreciate any feedback or help given, my apologies for the long post.

I also have another question (which may be related to this?) I don't quite know how you break up into and but that's not in an exercise so I'm happy for this to go unanswered

The definition of is not clear from the question. I could think of two possible interpretations: is a set consisting of any n natural numbers, or another interpretation is that consists of numbers from 1 to n.

In the latter interpretation, it is pretty straightforward to prove that the union of all such will essentially be N

which has an injective mapping from

In the former case I am confused if it can be shown that there will be a injection from to the union because I can always select arbitrary which always excludes certain numbers from (say none of the has 1) then the union will essentially be . In this case it is easy to show a surjection from to but will it also be an injection because of the their infinite cardinality?

I have submitted an erratum for this but for ex 8.2.3 I was wondering where it is intended to prove that the series diverges? Is it for a->infinity? (going by the notation of the definition in ex. 8.1.1)

If so I have been noodling around on this one for a while and I can't figure out how to prove it? I've tried via directly solving for k given M, but you just end up with a big mess of an equation that is hard to solve. Trying to show that the derivative goes to infinity as k goes to infinity (and thus that a_n grows without bound) also results in a big mess.

I know intuitively that the exponential in the numerator will eventually 'grow faster' than the polynomial in the denominator, but exactly how to prove this eludes me.

I am interested what the thought process was to write the polynomial "in a strange way"? So had a google and found your article: https://jeremykun.com/2014/06/23/the-mathematics-of-secret-sharing/

Where you write the above "in a slightly different way"

Is this form a common form for mathematicians something basic I should know about?

It seems like it is related too the Lagrange Theorem?

I liked your book and I worked along the chapter about polynomials as recommended: Making up examples along the way and making sure I understood the formulas.
Now I have gotten to the exercises for this chapter.

I don't know how to even try and solve exercises 2.3 and 2.4

Either I am overlooking something or the road to these exercises did not in the least equip me with any of the tools I would need to tackle them. I was motivated enough by the beginning of "A programmers introduction to mathematics" to dedicate an hour each day learning math. The exercises make me reevaluate this decision. Who is this book for? Someone already well versed in mathematical proofs? If not, how am I supposed to develop these skills with the exercises given?

For the live of me I don't know why mathematicians do this... The chapters are challenging but understandable, the first couple of exercises are really easy and then BAM: "YOU THOUGHT YOU WERE ABLE TO DO MATH? YOU THOUGHT WROOOOONG!!!"

I noticed this pattern in other math books: It is like a beginners guide to programming going over basic control structures in the first chapter and then asking about the difference between abstract base classes and interfaces in the exercises.

So, after the venting, as someone who really liked everything up to the exercises: What should I do here?

Regarding the following sentence at page 26: "Moreover, for each choice of y you get a different interpolating polynomial (this is due to Theorem 2.3)".

Why is this due to Theorem 2.3 (zero polynomial) and not to Theorem 2.2? I can't find any reference to the zero polynomial in the first claim.
For the Theorem 2.2, given a list of d + 1 points and choosing the y of one of those points don't we just get a different input list that generates a different interpolating polynomial?

I just want to ensure if my reasoning w.r.t. the examples I thought of for countable sets is correct or not.

Example 1:

In this case we can write a surjection F: N -> A as follows:

Example 2:

A = {set of all positive even integers}
In this case we could write a surjection such as:
F = {(0, 0), (1, 0), (2, 2), (3, 2), (4, 4), (5, 4),...}

On the other hand if we define A as

Then it is not possible to define a surjection F: N -> A (although I am not sure how to prove this).
Therefore A is not countable.

I am wondering what is the implication if the function F: N -> A is bijective. Does that have any special meaning?

Not really sure how to approach this proof. Help would be greatly appreciated :)

Is there a pacing guide for the book perchance?

I'm currently in the same mind set as #13.

I have no previous mathematical experience. I was happily reading along the first chapter excited at the presentation and finally understanding dots you were connecting. Right up until the very top of page 16.
"If we solve for a₀ in the first equation, we get a₀ = 3 - 2a₁" and all I could think was "do we?". I've re-read this section on three separate days now and I can't find any path that would lead me to this result based on information given, let alone everything that comes after. At best I might have thought: a₀ + a₁ * 2 = 3 could solve to a₀ = 1, a₁ = 1, 1 + 1 * 2 = 3. The simplest possible calculation I know how to make, and it would obviously fall apart for the second equation.

In #13 you mention we don't need an understanding of every concept before moving on, which goes against page 10 that explicitly says "the unspoken rule is that the reader should not continue unless the reader understands what the definition is saying". So I'm not sure what to feel. Is the expectation I already have an understanding of this, or did I clearly overlook an explanation somewhere?

By definition n and m are the largest degree in each respective polynomial, which means the the terms a_nx^n and b_mx^m are the the largest terms of each respective polynomial, so the largest the degree will be is (a_nx^n )(b_mx^m) or a_nb_mx^(n + m). How can I show that these are the largest, is there a step in relation to the previous question?

How to "prove it using elementary means" and show that "there must be some input on which they disagree"?

Since there are exactly n preferences in the list and exactly n Suitors, we get the bound a_t <= n² (each Suitor could be at the very end of their list; come up with an example to show this can happen!)

I spent way too long trying to write out such an example and gave up, and went searching for examples. According to a few search results, the worst case appears to actually be n² - 2n + 2, which I won't attempt to prove, since I'm not very practiced at maths, which is why I'm reading this book.

source

Exercise 4.2 (page 59) is unclear: Prove De Morgans law for sets (A∩B)^C ? A^C∪B^C. "C" is never defined, and as the preceding exercise is on power sets, this suggests that "x^C" should be read as a power set with the set C, not as the set complement.

I'm stuck on exercise 14-17, and I don't know how to proceed. This exercise involves transforming the computational graph to be able to detect digits from 0 to 9, not just 0 and 1. I converted the labels to a 'one hot' format, created the SoftMax node and CrossEntropyErrorNode, but I can't get it to work. Can you give me some hints to solve it. Here I attach the 2 files.

acm_neural_network_softmax.zip

Thank you in advance!

What is wrong with the following code for compute_global_param_gradient --

def compute_global_param_grad(self):
        # dE/dw_i = dE/df*df/dw_i 
        return [
            self.global_grad * self.local_param_grad[i]
            for i in range(arglen))
        ]

The author shows how to compute the number of games in a single-elimination tournament, using the following notation:
X - set of games
Y - set of players
L - set of losers (players that lost games)
f: X -> Y - function that given a game, returns its loser (injection - each game has unique loser)

Next, the author claims that in the case of a double-elimination tournament:

you won’t have an injection, but a so-called “double-cover” of the set of players. What I mean by double-cover is that every y ∈ Y has a preimage f^{−1}(y) = {x ∈ X : f(x) = y} of size exactly 2

However, isn't this statement false? If Y is the set of ALL players (losers and a single winner), then all, but one of the elements of Y will have a preimage of size exactly 2, while the winner will have an preimage of size 0 or 1. Am I missing something or did the author make a mistake or mean to write L instead of Y in the cited fragment?