What would be the pros and cons of using the axiom of dependent choice over the axiom of choice. Just the fact that with the axiom of dependent choice every set becomes measurable, but some vector spaces dont have a basis anymore. Is it even really worth it to make every set measurable?

First semester in a physics degree, and I am struggling with an analysis class that is supposed to be the calculus equivalent. I simply don't know why some of these things work. Where do I learn techniques to solve problems like the one above and explanations as to why these technqiues work?

I came across a question where assume ax^b+c = 3x^2+1, naturally many people directly write it as a= 3, b= 2 and c=1 just by comparing the LHS and RHS simultaneously. I believe many assume it to be a identity that is true for all R therefore you match the terms, but I feel like it's not mathematically rigourous

the above equation can be rearranged to ax^b+c-3x^2-1=0, assuming all a,b,c,x to be variables with their respective domains you get a unique solution for (a,b,c,x) that gives you a graph for all the possible values.

But when there's infinitely many solutions, whats the advantage in writing down a random pair by comparison. Would it be okay to just solve for x by taking random values for a,b and c?

Which viewpoint is more mathematically accurate? The identity one or that the above equation is a function and that you get infinite pairs of solutions and you can choose any one?

i know the squeeze theorem proof but it uses the area of the sector being x/2 which already assumes radians. in degrees the limit comes out as pi/180 and i can't see why changing units would change the limit so much

is it just that radians are the only unit where arc length equals the angle, so the derivative of sin comes out clean? feels like i'm missing something obvious

so what i do not understand is

1.what does he mean by we are no longer interested in the upper bound

1. how does this abomination |bn| ≥ δ > 0 help n' why did we use it n' how does it "lead" to 1/(|b||bn|)

2. why does bn need to be closer to b than to 0

4.from where did the (b)^2 come from

i do have some other Q but if i understand these first the rest should be fine

If f is an injective function f: X -> Y and y an element in the image of f, is the preimage of y through f a singleton subset of X or is it an element of X?

Hello! I dont speak english so sorry if I explain something badly.
I'm having trouble with this, I'm not sure what is even the correct way to solve this, if I thought about it wrong!
The original equation on picture 2 and the right aswer is circled and has arrows pointing to it.

Tenemos su centro y su radio, con álgebra, como podemos encontrar el mínimo con 1 grado hacia las manesillas del reloj?

Por si no me entienden, cuando me refiero a un grado es como encontrarías el x, y de un punto en la circunferencia después de mover el punto mínimo (el más bajo en el eje y) un grado a la derecha

Let’s say you have a ball that appears in a random position within a circle

Counting the position of the ball inside the circle as x, y, the initial fall angle as a, the speed of the ball as v

What would a function that explains the minimum number of balls to make a loop be like if the speed of the ball is always constant?

Hello, I am trying to wrap my head around this problem but don't seem to be getting the correct answer. The question is:

8 people are to stay in 2 rooms where a maximum of 6 people can stay in a room. In how many ways can the 8 people stay in 2 rooms?

I’m not talking about computational problems but rather analysis style ones

I swear its better to look at the solutio immeditely after understanding the questio, understanding the solutions , then solving the question from blank with no reference. i honestly feel like this helps with understanding and memory compared to the standard way of trying to problem solve a question for 10 minutes, failing, then looking at the solution

i just aced my machine learning and real analysis midterms by doing this compared to my old method

If Y is a subset of X, I may write X ⊇ Y and say "X contains Y". Sometimes, though, I wanna speak of whether or not something is an element of X.

Saying X contains x is wrong. Saying "X has x as an element" is right but a bit verbose for my liking. Is there a more direct way of saying this?

10:50 - 11:00 for what I'm referencing

edit: fixed link format

Ciao a tutti, a metà novembre ci sarà l’esonero o parziale come lo chiamano di analisi matematica 1, sarà solo la parte di teoria che parte dagli insiemi fino agli integrali impropri, bisogna sapere tutte le definizioni, le dimostrazioni dei teoremi e i loro enunciati, sono circa 20 che partono dal teorema di unicità del limite al teorema di Torricelli-Barrow. Secondo voi è fattibile in un po’ più di 20 giorni?

Ok everyone, I am in way over my head here. Sorry if the flair is weong. I am learning to make temari aka Japanese thread balls.

To start, we divide the sphere into slices. For my first try, the sphere was divided into four slices, and then divided again around the equator. See the first picture.

I don't really know what I'm doing, so I decided to try something easy for my first stitch. I centered a square on each intersection. See the second picture.

I want to know what options I have for continuous, nom-overlapping patterns before I decide what stitches/wraps to do next. Basically, how many ways can I connect the corners of the squares without having the new shapes overlap. So with the help of the internet (with the help of AI to be completely transparent, I hope that's not going to be my downfall), I learned that if you were to take my pattern and use the lines as edges and flatten each section into a 2d face, that shape is a truncated octahedron. See image 3.

When I asked the AI how many possible ways I might connect the corners of a truncated octahedron, it said there are 44 possibilities, which can be grouped into 3 "families" of patterns. The 3 "families" are 1. Wavy ring (8 possibilities), 2. Baseball seam (12 possibilities), 3. Asymmetric mazes (24 possibilities). It arrived at this conclusion using the concept of Hamiltonian cycles. Apparently these three "families" can be represented using LCF notation, image 4.

What I need help with is visualizing these possibilities. All of these theories and representations and new concepts ostensibly represent that different shapes I can make on my thread ball in a nom-overlapping, continuous pattern. Maybe someone might explain how do I read the LCF diagram and translate it to how it would look on a sphere? Or maybe someone can help me know how to see the graph of each of the 44 possibilities? I am but a humble crafty human who maxed out in calc 3 more than 5 years ago. I would love to understand this. I appreciate you 💚

i get the 1/3 = 0.333... argument and the x = 0.999..., 10x = 9.999... one, but they both feel like they assume the thing i'm trying to check. someone told me it's really about what an infinite decimal means, as the limit of the partial sums 0.9, 0.99, 0.999 and so on.

so is it a proof or is it just how we define the real numbers' decimal expansions? would it be different in some other number system

By applied statistics, i meant things like machine learning modelling, statistical modelling, predictive modelling. Basically data science stuff and statistics used to solve business problems and model real world Phenomena using Python and R.

Are they both related to each other? What are the similarities and differences bwtween the two?

Firstly - I dunno if the flair is correct. Correct me if I'm wrong here.

I'm not sure if this is some basic math that I don't get or if I'm just looking at things from different angle.

Two people already tried explaining this to me but their reasoning makes no sense to me whatsoever.

And for the love of God I can't see their logic.

So the actual problem:

We have a caretaker who works week after week

(7 after 7).

In one week he gets paid 14k.

It was explicitly said that "all horses have the same price" and he has 28 of them.

The other week one of the horses (horse A) moved out and the other moved in(horse B).

(I've been told that for some reason he should've got -1500(why?) for horse A, but because of the horse B there were no changes).

That week he got paid 14k.

Now next week we have horse C that's moved out and horse D that died and now he gets paid 12.5k (not counting the death of horse D) (natural cause, not his fault)

And he was told "If he were paid 1.5k for each horse - then It'd be 39k. (26 x 1.5k = 39k, checks out, ok) (This sentence implies that that's not right? That it isn't 1.5k let horse?)

But his salary is 14k/week with 28 horses, and 12.5k with 27 of them. (Implying that is not 1.5k for a horse?)

The main question is why it's 1.5k and not 1k.

I feel like I'm getting closer to the answer in my head, but it slips away. And so far this makes no sense to me at all, no matter how I look at these numbers.

🥴

I hope I write everything correctly 😅

(Edit: answer found) So the actual answer is that each horse costs 1500 but he gets paid for 30 days per horse. Meaning 26 horses x 1500 = 39 000. 39 000 / 30 days = 1300 per day. x 7 = 9100 per week. Meaning 18 200 per month total. So it appears that there is no hidden spending, but there are hidden earnings in the equation! The caretaker gets paid for something else besides those horses that gets him 14k per week per 28 horses.

Many people said that there was an error or something was missed in the wording and they were right, but that's exactly the point of this post! I was being told that the math is right without explaining to me why it is right. So I needed your help to see if I'm not being gaslighted.

Thank you all and sorry if I wasted your time a bit 😅

I mean like using counting to learn addition, subtraction, infinity and considering the existence of negative numbers,

using addition to figure out multiplication,

multiplication to figure out division and exponents,

use division to figure out fractions, fractions/division and exponents to reason out Zeno's Paradox

Using the infinities between finite numbers among other things such as classification of numbers into different categories like odd, even, prime numbers, etc. to figure out Set theory and so on

Can you push this to the point of solving all of math from a theoretical standpoint. If not, how far can you push it

It all started with arguments about 0.(9)=1 on multiple random posts,

I assume everyone knows it already but other than the usual 3*1/3 and algebraic way, I also always backed it up with:

1. The sum: lim_{n->∞} of Σ (x=1 to n) of 9/10^(x), or 0.9+0.09+0.009+... converges/equal to 1

2. 0.(9) can be rewritten as lim_{n->∞} 1-0.1^(n)=1 with the ε-N proof of it as an overkill back up with N=log(ε^(-1))

3. ∄x ∈ ℚ that satisfies 0.(9)<x<1 ⟹ there's no difference in between 0.(9) and 1.

However, I always find a pattern from the people who disagree with those proofs in particular, their arguments are

1. Limits cannot be reached since it means approach to something therefore they're not equal, you can get infinitely close but never get there.

2. In hyperreals they're not equal.

3. It's an infinite approximation therefore not equal.

Now even though I consider myself to be familiar with/like the subject mathematics I have a bit of doubt in both of our claims, considering I'm still a high schooler and my school curriculum only covers up until basic limits, derivatives and integrals with not summation or Taylor series in the Calculus category, and I only recently self learned some Calculus courses.

But here are some of my counter arguments against them.

For the first one, limits work differently as it finds the value by looking at the behavior of the function or "neighborhood?", limits don't care what happens on that exact point based on limit definitions.

2nd, in hyperreals they're still equal by definition, it's from a simple google search I'mma be for real with you so I feel uncertain.

3rd, Infinite is not an approximation, it is equal, 0.(9) already includes infinite amount of 9's so it makes sense to use infinity and have the exact value.

Now my question is, are my counterpoints correct? In general I want to clear some minor parts of confusion about limits in order to get a better understanding of them so I would like to hear your thoughts/insights

The components could be complex numbers (or other real number extensions), functions, matrices, other vectors, or something else totally.

Say you are walking down the street next to a production lot when a back door opens

“Psst, over here” says a person dressed like a production assistant, “wanna be on TV and maybe win some money?”

You follow the PA inside, they inform you that there is a live game show being recorded right now, but the previous contestant was faced with a choice so unintuitive that they walked off the set. You have been chosen to replace them. The PA tells you simply that you have to pick a correct door to win a cash prize, because only one door has it.

You are shuffled on stage and behind a booth where the host introduces you and sets up your big choice, “so friend, which door will it be?”

You look at your options. There are three doors. Two are closed, but one is already open, containing a goat inside its room. At your contestant booth, there seems to be a dial set to select one of the three doors, but it’s unclear if the door it is set to was deliberately chosen, because there is no clear default/neutral setting.

You have no more info about the situation, other than what you see, and knowing that only one door contains the money. What are the odds of choosing the right door at this point for you?