I'm in year nine ( NI ) and want to study maths more to do my GCSEs sooner, are there any good resources to help me?

Let f : X → Y be a bijective function between two sets X and Y .

Show that there exists a bijection between X × X and X × Y .

I struggle with proving in relation to injective, surjective concepts ALOT but i feel like this proof is really trivial.. How would you approach this? my idea is a case proof where we first proof 1) it's injective and 2) it's surjective, but i have no idea how to do either of those things

Thanks and have a great day !

I havent done much math for the past 5 years, I'm 21 now. I noticed that I became a little stupid after years of not really using my brain that much and only consuming content, and this vexes me.

I probably forgot a large chunk of what I learned in high school, and I'm wondering how I can get started again? Do you have any advice on that?

I’m disabled, and on the waitlist for Section 8 housing which in my neck of the woods can take years. I started out at 13,305 on the waitlist on 3/12/24 and have a series of data points (listed below) of where I was at on the waitlist on various dates. I was wondering if it would be possible to predict, based on the average amount of time passed and difference between the numbers, when I would get to 0 on the waitlist (the end goal) and how I would do that mathematically. Would I need a graph? Thanks, math friends!!

(13,305-3/12/24) (11,809-3/21/24) (11,783-3/30/24) (11,685-4/16/24) (11,649-4/19/24) (11,618-4/29/24) (11,565-5/15/24) (11,420-6/7/24) (11,307-7/5/24) (11,297-8/6/24) (11,200-8/23/24) (11,149-9/5/24) (11,105-9/26/24) (11,020-10/11/24) (10,885-11/8/24) (10,871-11/19/24) (10,841-12/7/24) (10,835-12/21/24) (10,830-01/4/25) (10,829-01/12/25) (10,826-02/1/25)

https://en.wikipedia.org/wiki/Boolean_algebra#Basic_operations

x ∧ y = 1 if x = y = 1, x ∧ y = 0 otherwise

I am confused with this bit: if x = y = 1

What is this saying? What I get is, it that x ∧ y will be = 1 if x is = y, but I am confused at the = 1 at the end here. What is this saying? (Sorry if this is a really basic problem, I am kinda bad at math). I've been staring at it for like 5 minutes and cannot figure out what this means.

A question abt numerical series U(0)=0 U(n+1)=4-U(n) The question is prouve that U(2n) is arithmetic

As we know, area is calculated by multiplying length by width. If someone asked why is that, and why do you call it square area? you would tell him "well, imagine a square, you have 3 rows, and 3 columns with squares, and each little square equals 1 square unit".Now think of it that way - You are the person that is just inventing the idea of area, how could you know that the area of the little square is going to be called 1 square unit, and why would you call it like that, as you are just trying to create the definition for it by decomposing a larger square by counting the little squares inside of it?

If I have a right angle triangle (in cross section) with area A, and wanted to revolve the triangle around a central axis at some distance (X) away from the triangle, what would I multiply the area by to get the volume of the new shape? Imagine the 3D shape looking like a donut/bagel that tapers off to a sharp point at the base. Should I just multiply A by the circumference of the circle with radius + distance to midpoint of triangle (X+1/3 base of triangle)?

Recently, I appeared for an Olympiad wherein there was a question that stated: "Prove that a^3/b + b^3/c + c^3/a >= ab + bc + ca for all real nos. a,b,c > 0" Now I tried many things like factoring out common terms or AM-GM inequality and just about anything I knew about inequalities yet I was unable to prove the question. What would be the thought process and the method/theorems used to prove this? Also will this apply to complex numbers?

I recently appeared for an Olympiad wherein I had the following question: "Find the Maximum possible value of K such that the sum of K consecutive positive integers(not necessarily starting from 1) is 2025." My answer came out to be >!50!<, I solved with some pretty lofty logic, like comparing coefficients and also just using the hit and trial method. Also is there a generalized approach for this? If possible, please give the sequence.

I am an electronics and telecomunication engineering student, I like math, physics and especially astronomy, do you have any recomandation of books, books/materials I can use to self-study all of these subjects from ground up? I don`t fell like we are doing enough of these in uni, and the math i fell like it just enough to cover the engineering part and a lot corners are cut. As much as I enjoy what I study in uni, I still dream on working in one of these fields. Thanks in advance! :)

https://preview.redd.it/gap-in-a-smooth-line-does-not-impact-its-existence-of-limit-v0-6u1q3ufs8oge1.jpg?width=496&format=pjpg&auto=webp&s=97742bb5204189dd6559593534d8ffb2f757a551

To my little understanding, the reason why a gap in a smooth line does not impacts its existence of limit in that point is that we are concerned in the values around the point and not exactly the point. This then is the sole determinant of the existence of a limit in a point: the values around the point on both sides matter but not the value exactly in the point. Apart from confirming the existence of limit, is there any specific reason for this as it can be misleading because x not equal to 1 in the example when f(x) = 3 yet we are declaring that as x tends to 1, y tends to 3.

how does one become good at knowing how to solve a problem? i know part of the answer is practice, but what kind of practice is needed? like, when i do amc10/aime/usaco, i waste so much time trying to figure out how to approach a problem that i just give up and move on so that i dont run out of time. how do i fix that? am i practicing wrong?

Hi guys, sorry for the stupid question. How does -(1/r)=ln(x)-1/2 become r=2/(-2ln(x)+1) ? Thanks.

A 9-foot pole is set into the ground 30 feet from the base of an 18-foot pole, and the cables are run from the top of each pole to the base of the other pole. How many feet from the base of the taller pole is the point on the ground directly under where the cables cross?

Solution (I added letters to each of the points)

I really don't understand this solution, how are any of the triangles similar? I would really appreciate it if anyone could tell me what two triangles are similar and why.

I’m a NEET with lots of free time, so I dedicate myself to learning. I’ve taught myself philosophy, classical studies, and I love the humanities. However, I have dyscalculia and struggle with math. I really wish I could learn it I find it a beautiful subject, and mastering it would not only challenge me but also be a great way to pass the time.

How can I start? Could you give me a detailed plan to improve, starting from basic math and progressing to advanced levels? I’d love a structured approach that helps me build a solid foundation.

Hi everyone;

I have what I believe is a simple math question, that I can’t wrap my head around. If someone could please tell me in plain English how the math equation works, it would solve a mystery for me. Thanks in advance!

A marathon runner starts at 100 yards from the finish line. At some point they are 50 yards (explained mathematically as 100/2 = 50) then at a later point that are 25 yards from the finish line (mathematically 50/2 =25). On paper, as the runner moves towards the finish line, they would be at the point found by dividing the remaining distance by 2 over and over.

My question is: On PAPER (figuratively), as you keep dividing by 2, you would keep letting a smaller and smaller number with infinite decimal possibilities, but in REAL LIFE, the runner passes the finish line, and is not stoped infinitely before ever reaching it. How can this be explained in the math equation?

Thank you for explaining this to me as I have wondered about it for over 30 years 😄

This is probably a very picky question, but is the singleton {s} the same as just normal s? Similarly, is the one by one matrix [x] the same as the scalar x?

I was watching a Flammable Maths video. It was about the gamma function witch derived the Euler's Reflection formula with an ugly integral. Before I tried to came up with the solution myself, and after that I watched his video. It was very similar to my solution but he didn't prove this relationship, and I'm stuck. Are there any paper, any proof, anything that makes this relationship even logical or real? Sorry if my English might sound a bit bad, but i'm tired as hell and it's my second language. the link to the identity I need to prove

I’m studying right now about evaluating trigonometric functions using reference angles. I could understand how they find the reference angle and then it would result with an answer like sin 45° or tan 120°. But I don’t get it how they convert it to numbers like √2/2 and √3. Could someone explain? I’m new to plane trigonometry and I’m stuck in this part.

I see that this is used in a lot of places but i dont know why or when can i use it, for example for the derivative when they say lim (f(x)-f(a))/(x-a) x--->a and lim (f(a+h)-f(a))/h h--->0 are equal, i know this is true when f is continuous, but i want to learn about the more general case

I've just started College Math in my freshman year, I'm doing a class called MAT-035: Concepts Of Algebra and I'm having a bit of trouble.

I'm learning, but not fast enough to keep up with the course. It's an 8-week course that is done on the computer while you are in class with a teacher walking around tudoring people. The website is called hawkeslearning and it's pretty barebones when it comes to teaching imo.

I don't really have an intutive grasp on how I'm supposed to solve algebra equations, especially because I did not do high school and just recently got my HiSET equivalent.

The important details of how to solve some of these equations were not explained to me, and took me too long to figure out. For instance: The solution was explained, but not that the whole idea of these particular problems was moving the variables to the left and the constants to the right by doing the opposite operation on terms to get 0.

I have also had trouble understanding what terms I'm supposed to start adding/subtracting on both sides first and in what order.

Nothing has been explained to me to help me intuitively understand how to solve the questions. I'm able to comprehend the material, but not understand the "why" of the solutions.

Is there any rescources I could use for this? Or any advice?

I haven’t done math in 7 years and now I’m in college algebra and struggling hard. I thought I was understanding things but it turns out I don’t. Why does X not ONLY equal 22 in the equation (x+5)^(2/3)=9?

My math experience is just how to prove It, so I'm a novice

According to my education, school books do not care to define precisely the objects they're talking about.

Take as an example euclidean geometry. The books I have read do not build the concepts. They want you to know some properties and to use them with guessing in order to solve problems

However I have started to read Hung Hsi wu books on elementary math and he makes the effort to build the concepts precisely. But this makes the subject more difficult than It would be without and makes me question if this Is deepening my understanding or not

As another example, consider fraction addition

One could memorize a/b+c/d=(ad+cb)/bd and do the exercise without caring about what rhs and lhs mean, as I have done in highschool.

Does really knowing what a/b+c/d means and how can I write It as rhs make my understanding more grounded?

Sometimes I feel that being rigorous,i.e. trying to understand what one is talking about is a loss of time. Books are exercise oriented. They want you to be good at using equation and rules they don't care about what they're talking about so much

What's the approach of a math student? Do you always make sure you know what you're talking about? If the books Is not rigorous enough are you able to build the theory on your own?

Or do you just accept that maybe the author goal Is to make you memorize formulae and do as required?

A box contains 30 red balls, 30 white balls, and 30 blue balls. If 10 balls are selected at random, without replacement, what is the probability that at least one color will be missing from the selection?

I'm thinking: P(at least one color missing) = 1 - P(all colors present)

And P(all colors present) = (30 choose 1)^3 * (87 chose 7) divided by (90 choose 10)

This is, however, larger than 1. So something is really off here.

As you can see, I tried to go about this by using combinatorics. However, it seems that the inclusision-exclusion principle is needed. I don't understand at all, why this is necessary, however.

I have been off maths for a long time but I want to get my skills back in maths like quickly solving calculus and algebra

Hey guys, Im at the series part in Understanding Analysis by Abott, which btw is probably one of the better analysis books out there(I literally gave up with Tao's one).

Even still, it takes me like 1-2 hours trying to understand the section, probably like 30min - 2 hours (anything longer than that I give up and look answers) for most of the exercises. It takes me on average like 2 days to barely get past a section even tho the it is like 5-6 pages with exercise included.

Is there a way to efficiently study this? At this rate, itll probably take me about half a year to complete Understanding Analysis. I don't mind but I would prefer optimizing the process.

I suck at math so can someone please explain how to calculate this?

The odds of getting a thing are 1/1m but i also have 700% luck so therefore 1/1m is 0.0001% and then i multiply by 700% so 0.000007 = 0.0007%.

Is that correct way to approach it?