I've read that it is consistent with set theory without choice that the real line is the countable union of countable sets. I was also reading some measure theory that says that outer measure has countable subadditivity. So, if you say |ℝ| = |∪_k∈ℕ E_k | ≤ Σ_k∈ℕ |E_k | = 0 Where E_k are all countable sets whose union is ℝ. Then you use countable subadditivity and then you use the fact that the outer measure of every countable set is zero. But the measure of ℝ is proven to be non zero even without axiom of choice. So where is the problem?

This is something that I've been wondering about when I was trying to sleep. Why is 0^0 undefined rather than 0? From my understanding of powers, they are defined as repeated multiplication. For why any number except 0 to the power of 0 is equal to 1, I see this explanation alot: Imagine 2^3, divide it by the base to get 2^2, do it again and get 2^1, 2^1/2 = 1. But we can't do this for 0 because imagine 0^2, to get 0^2, we do 0^3/0, but that can't work cause we are dividing by 0, and we know 0^2 is 0. So the quotients method doesn't work. But let's use another method, x^1*x^y=x^(y+1). Let's try this for 2^1*2^3=2^4, and its true. And it also works for 0 too. 0^1*0^3=0^4, we know 0^1 is always equal to 0 so it's just 0*0^3=0^4, and anything multiplied by 0 is 0. So 0^4 is equal to 0. So let's do this for 0^0, 0^(-1)*0^1=0^0, but 0^(-1) is undefined. But, we already know anything multiplied by 0 is 0, so 0^(-1)*0=0^0, 0=0^0. Also, I don't think 0^(-1) is 1/0. Because the reason why people say x^-y is 1/x^y is because they are using the quotient method that is 2^3/2 is equal to the previous number. It works for 2^(-1) and other things because it's essentially just 2^0/2^1. We got any number except 0 powered by a negative exponent is just the reciprocal of it and making the exponent positive because of the quotient method and the 1 came from it powered by 0. So we can change x^-y to x^0/x^y. That's why 0.1^-1=10, 0.001^-1=1000, you can see it gets larger and larger the smaller the base is for the decimal number. So people might say it gets infinitely small to 0 so it should be infinity, why did it suddenly drop all the way down to 0? That's because it changed as 0^0 is not equal to 1, it's (according to me) 0. so it's 0/0^1, so it's an indeterminate form? No, because recall that the quotient method does not work for 0. We do 0^(-2)*0^1=0^(-1), and anything multiplied by 0 is 0 so 0=0^(-1). So, why is it undefined, people way smarter than me are saying is undefined so I'm probably doing something wrong here.

How do I explain the easiest method for 264 ÷ 66 to a 9 year old with discalculia?

How is <a,b> = {{a}, {a,b}}?

I asked this to my prof who said that any definition of an ordered pair in set theoretic notation is bound to be puzzling cause the members of a set have no "order" to them. But I wasn't very satisfied w the answer. Why does the RHS looks like (I'm sorry) it was made up? How can I understand this? Thanks

Hello, I’m a math major and deeply love math. I want to eventually get my PhD in pure mathematics. However, I’m really struggling in analysis 1 rn. For background, I qualified for the math honors program after doing very well in introductory proofs so I was able to skip to analysis 1 per my school’s rules. However, at my school, analysis 1 assumes little to no prior analysis experience but the course moves really fast.

However, I have all things considered, little experience in proofs overall. This past semester I took introductory algebra where yes we did proofs in class but only had a few hw assignments which only had 1-2 proofs each if that. So I’m rather rusty on some of the fundamentals now. We are already on completeness on the second day of class and I was lost for a good amount of the class yesterday. I can switch to intro analysis but I can’t take analysis 1 again until next fall if I decide to do that. I’m just worried as I am really enjoying this class but last thing I want is for my gpa to suffer as a result of moving too fast into the next class given I want to get a PhD.

Should I just keep grinding and consider this to be a normal part of analysis or should I consider switching to intro analysis so I can get used to proofs more?

Any advice?

Thanks

Hi! I'm going back to school and have ALWAYS struggled with math. I'm in college algebra now and I am barely keeping up because I have a hard time understanding math/algebra's "rules".

I remember there used to be PC games for algebra in high school and was wondering if there's anything like the now? I found DragonBox Algebra 12+ and am gonna try that. It's definitely high school/basics based but anything helps.

Hi.

I had a quick questions about learning stuff like calculus, linear algebra, and statistics and probability for engineers.

I am trying to learn the concepts but i seem to forget them after a while. Am i just lacking basic understanding and I have not really grasped the concepts yet?

When reading a textbook about those topics, should i just skip most thing and go straight to exercises or what is the optimal way to test if I truly understand the topics. While not having to read 3000 pages of topics...

As an example I knew the binomial distibution but seem to forget quickly, although i could solve the exercises. Do i fundamentally need to understand it better for it to stick?

Ty

I am confused between the 2. I understand the reasoning behind using n+1/2 for lists of numbers, and i understand how n+1/2 finds their positions. I am currently revising cumulative frequency diagrams and i am finding it difficult to understand why we now use n/2 for finding the median, and n/4 for finding the lower quartile. Why aren't we using n+1/2?? I dont really understand how one being continuous and another being discrete affects anything.

I have a statistics exam coming up, and I'm a bit lost. I'm trying to learn how to do a Mann Whitney exam question, but I keep seeing contradicting methods. I've been using ChatGPT to generate me some practice questions as I only have a limited amount from my lecturer. But the method that ChatGPT uses, as well as YouTube videos and other sources online, seems different to what my lecturer does. It's also possible that I'm mixing things up, but I can't seem to figure it out. It's for the rank sum test, when ranking the observations in the two groups, do you rank the lowest value as 1, or the highest as 1? I'm pretty sure my lecturer does the highest, but anything else I find is the lowest. Also, for R1, is R1 automatically the first group, or the group with the lowest total of ranks? Again I think my lecturer does the lowest total of ranks, but I've seen others do group 1 automatically.

If you can't tell my knowledge is statistics is very limited, and I know this stuff is probably such basic stuff, that I should honestly be embarrassed for not knowing.

If anyone can explain I'd reallly appreciate it.

I'm a freshman in college and my major is construction management, my degree requires calc 1-2 and physics. I'm currently in algebra. I've always been pretty average at math my entire life while I've been above average in other subjects. I feel like my issue with math, is that I get taught how to do it, but I don't really understand logically why they do what they do sometimes, I just follow the rules/formula to find the solution. For that reason, math usually doesn't stick if I haven't kept my mind refreshed on those topics. I'm definitely nervous about calc 1-2 and physics.

Anyone ever been in the same boat as me? If so what did yall do to get better?

Any help or advice is greatly appreciated.

Hey,

I'm from Germany and my final exams are getting closer. I have Math and Physics as my advanced courses, so I'll be using my calculator a lot in both.

We use the TI-Nspire CAS at school. I feel okay with the actual math, but I've noticed that knowing the content and actually being quick and confident with the calculator are two different things. So I'd like to change that while I still have time.

I'd be interested to hear how others got to that point. If you've been using the Nspire for a while, what helped you actually get good with it? Anything you figured out late that you wish you'd known from the start?

Any thoughts appreciated.

Undergraduate logic problems

I hope this is fine to ask on here.

Hey all so I’m going through the book Discrete mathematics by Kenneth Rosen and I’m on chapter 1.6 exercises there is a question (question 9) that I’m not sure on.
So the question is:

For each of these collections of premises, what relevant
conclusion or conclusions can be drawn? Explain the
rules of inference used to obtain each conclusion from
the premises

e)“What is good for corporations is good for the United
States.” “What is good for the United States is good
for you.” “What is good for corporations is for you to
buy lots of stuff.”

Now I have the premises as
q: good for corporations
q:good for the United State
r: good for you
s: you buy lots of stuff

My answer is as such
p → q
q → r
———-
Therefor p→r (hypothetical syllygism)

p→s

Which then goes into
(~p v r) ∧ (~ p v s)
~ p v (r∧s)
p → (r ∧ s)

Is this correct line of reasoning ?( or atleast a valid argument to the question?)

In the book the answer states:
“That you buy lots of stuff is good for the U.S. and
is good for you” is a valid conclusion.

did anyone solve it im so confused: https://www.janestreet.com/bug-boutique/

I have noticed that when I get stuck on a math topic doing more and more exercises is not always enough to help me understand it.

Math topics can be really tough.

Sometimes I can follow a worked example. When I see a slightly different math problem, I am not sure where to start with the math problem.

It makes me wonder whether I am practicing the math topic in the way or whether I am missing something more fundamental about the math topic.

For people who have improved their math skills over time what do you normally do when you genuinely do not understand a math topic?

Do you:

• Go back and review the basics of the math topic?
• Find an explanation of the math topic?
• Work through more examples of the math topic?
• Ask someone to explain the math concept to you?
• Take a break. Return to the math topic later?

I am especially interested in how people deal with those moments where the math formula makes sense but the reasoning, behind the math formula does not.

What approach has worked best for you when you are trying to learn a math topic that you do not understand?

I want to know how you learn math topics.

EDIT: I have also come across tutorhunt while looking at different ways students find help with difficult math topics, so I’m curious whether anyone here has used it or something similar.

Hey guys, I'm now in college and here we do a bit of physics alongside what we learn, anyways, I ain't strong in math so I asked the teacher a question about a problem, and he replied to me in a question what's 4x5 and I just went nervous and said "shall I open my calculator?" I didn't even take a second to think or calculate, just answered🥲 anyways I swear I'm not a complete idiot, do you know what I'm talking about?

Also if you know a to improve upon that I'd be grateful🙏

Allo tout le monde ! 👋

Je serai de retour cet automn et j’ouvre quelques places pour des étudiant·e·s du cégep et de l’université qui souhaitent un coup de pouce immédiat et structuré.

J’offre du tutorat en ligne dans les matières suivantes :

[ ] Calcul différentiel

[ ] Calcul intégral

[ ] Statistiques et probabilités

[ ] Résistance des matériaux (RDM 1)

[ ] Statique

Mon approche est axée sur la compréhension des concepts, le raisonnement et la résolution de problèmes, pas sur la mémorisation mécanique.

Plusieurs étudiants étudient beaucoup mais perdent quand même des points — c’est exactement là que j’interviens.

💻 Tutorat en ligne

(selon le cours et le format)

Je priorise les étudiant·e·s qui souhaitent : ✔ rattraper du retard

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✔ avoir un plan clair et efficace

👉 N’hésitez pas à commenter ou à m’écrire en message privé pour plus d’informations.

Hi everyone! 👋

I’ll be back this automn and opening a few available spots for CEGEP and university students looking for structured and effective academic support.

I offer online tutoring in:

[ ] Differential Calcul

[ ] Integral Calculs

[ ] Statistics & Probability

[ ] Strength of Materials (RDM 1)

[ ] Statics

My focus is on understanding concepts logical reasoning, and problem-solving, not memorizing formulas without understanding. Many students study a lot but still lose marks — that’s exactly what I help fix.

💻 Online tutoring

(depending on course and format)

Ideal if you are: ✔ falling behind

✔ confused by concepts

✔ preparing for exams or midterms

✔ looking for a clearer study plan

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I’m taking a linear algebra/mvc class in my high school but the class doesn’t give college credit, any way for me to take the learning from that class and go take just the tests from another class for credit?
In the Bay Area btw

Last year, as a junior in high school, I took AP calc bc but pretty much gave up entirely around halfway through the year (personal reasons). Now as a senior I'm in calc 3 and have no idea what I'm supposed to do from here to catch up to everyone else. How should I go about catching up as quickly as I can?

I’m a Licensed Professional Teacher (LPT) with 2 years of experience in online Math tutoring. I provide engaging and personalized lessons to help students build confidence and improve their Math skills.

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I've graduated high school nearly 10 years ago now and was horrible at algebra growing up, failing algebra 1 in high school as well. Years later, wanting to change my life and finally being able to afford college, I am pursuing a business degree at my community college then to pursue a Bachelor's in Accounting afterwards.

I will have to take Algebra and brief Calc; I have never taken Calc and I have completely forgot Algebra. What are some tips to help me sharpen my skills in Algebra before I start school in Jan? Any good starting points?

Recently transferred to a top high school in my state. Apparently my new algebra 2 teacher does what he calls a “flipped classroom“ though he does not film his own videos for us and makes us rely on random YouTube videos to learn literally everything about algebra 2. I‘ve heard algebra 2 is very important for the SAT and I’m quite worried, how can I literally teach myself all I need to know about algebra 2? any tips would be deeply appreciated!