I encountered a theorem which says: "every subspace of a separable space is separable". What if I pick a finite set? To my understanding a finite set is not countable as there's no bijection between a finite set and naturals.

On the X and Y axis, if i keep drawing a 5cm line to connect X and Y, for all values of X and Y below 5, i seem to be drawing a part of a circle.

I thought it’s a quarter of a circle with the center of the circle at 5,5 but it’s not (i used a drawing compass and the circle doesn’t match)

1)Where’s the center of this circle 2)if it’s not a quarter of a circle, what percentage of a circle it is

If you were to solve this task would you convert feet to inches to find k??? I'm just trying to grade myself. The official solution says k = 400 (because they used 12ft in the formula). I got k= 4800 because I converted ft to in. Would you consider my answer a mistake? How would you go if you encountered such problem. Since (c) is still correct I know that converting/not converting doesn't matter as long as you stick with how you calculate M. I thought that all the dimensions should be in sync.

Is there a solution to this simple riddle? Imagine any fraction a/b = x%. Where (a ± y)/(b ± y) = c/d. In this case c/d = (x ± y)%. That is, (a ± y)/(b ± y) = (x ± y)%. The last requirement is that a, b, c, d, x and y are numbers {R}. I don't know if this little riddle has a solution.

My strategy: (a+y)/(b+y)=c/d Hence, ad+yd=bc+yc…… y=(bc-ad)/(d-c)….. x%+y% = a/b + (bc-ad)/(d-c)100= ((ad-ac)100+b^{2c-abd)/(bd-bc)100.} But, this is not c/d??!!

I know the only possible answer this could be is observational study because the other 3 choices don't make sense here. But isn't this a survey and not an observational study? In an observational study, the person is observing the participants and collecting data without any contact with the participants. That is not the case here. Can someone who has more experience with statistics shed some light on this for me please?

Hi everyone, I came to this sub for the first time to ask this question that's been eating me up. The chat didn't explain it well, and there's already a test tomorrow.

Could anyone explain if the denominator would be 0+ or ​​0-, since x-x equals 0?

This would be necessary to determine if the result is + or - infinity.

The answer key for the question is - infinity, which implies that |x| - x is 0-, but why couldn't it be the other way around?

*The book is *O-Calculus-with-Analytic-Geometry-Leithold-Vol.-1

I get the idea here, but I think the proof has a hole. We established (pigeonhole principle) that no matter which radii you choose, there will always be at least one ball, which contains infinitely many terms. My issue is that it doesn't have to be always the same center x.

I am trying to rate a musical album from 1 out of 10. It has 12 songs. To figure out how many points each song has I divide the songs by 10 = 1,20 points each song. I liked 4 songs and half-liked two songs (so 0,60 points). So I did 4 × 1,20 + 0,6 + 0,6 = 6/10. But.. does this rating represent the fact I didn't like 5 whole songs? 5 × 1,20 + 0,6 + 0,6 = 7,20. 10 - 7,20 = 2,8/10. This is a different number.

I also noticed that if the 6/10 is made 6/14,4 which are the total number of points (12 × 1,20) and remove 4,4 from both numbers I get 2,8/10 too. Does this mean I shouldn't have done 12 ÷ 10 but 14,4 ÷ 10?

What I want to know is if there is a problem mathematically with all this and how to fairly rate the album in a way that represents exactly how many songs I liked and how many I disliked.

^{(I don't know what flair to use})

Would you just use pythagoras with the angles to get the total? Or can you just add them up?

This is a scatter plot where for a set of integers 1 to n, you find the number of odd numbers you encounter in the Collatz conjecture before reaching 1 (i.e. the number of times you apply 3n+1) and plot it on the x-axis. On the y-axis you find the largest power of 2 that divides n with no remainder and call it f, then you plot log(f*n) (for odd numbers f is just 1). The result is above.

There appears to be a rough mirror symmetry along a line of constant y which increases as the number of points you add increases. I can reason some features of the plot like why the line at x = 0 appears as it does but I can't reason why the overall behaviour.

I believe this question is equivalent to asking: why would the plots of log(f) and log(n) vs the number of odds look roughly like mirror images of each other, especially since plotting just f and just n vs the number of odds look completely different to each other?

So far, I have tried to find a relationship between log(f) and log(n) that explains this behaviour as well as the behaviour for other scatter plots with log(f*n) as an axis (since I think this could maybe be a more general behaviour not at all related to any chosen x-axis), but I have been unsuccessful.

Thank you.

so i got 7 as the top of the quarter circle because from left to right the length would be 13(8+5) and the base of the box on the right is 5 and we already have 1 number for the bottom, meaning the 2nd number would be 2. well looking at the whole thing we don’t need 6 of those numbers(the 3m on the right and 3m on the left) so naturally you subtract 6 from 13 and get 7. now what do i do from here to get the quarter circle. google has told me multiple things like the formula for a quarter circle is pi times radius of full circle squared and divided by 4. but in order for me to find the full circle i need the radius of the quarter circle, and to get the radius of the quarter circle i need to work backwards from the area. i literally cant do one without the other im so lost??

In school when you are taught how to round numbers, they tell you to round up when the next digit is from 0 to 4 and round down when the next digit is from 5 to 9. This seems a bit counter-intuitive at first because when the next digit is 5, shouldn't it just be exactly in the middle of the range and not at the top? For example 1.5 would be rounded to 2 rather than 1 but is it really closer to 2 than 1 or is it exactly the same distance? 1.51 is rounded to 2 using exactly the same logic by looking at the 5 after the 1 and rounding up and this time is it obviously closer to 2. But how about 1.5? Is it just rounded up because even though it is the centre, it still has to be rounded to either one of the values so it may as well be 2 because literally any other number with a 5 after the 1 would be closer to 2 so it makes the 'rule' easy to follow?

I know that i and -i share all properties or something like that but I don´t know how people figurated that out.
Is there some example that works for 1 and -1, but not for i and -i?

Hi everyone. I just started Alg 2/Trig honors and I am insanely confused. One of the questions on my HW really stumped me...

CRITICAL THINKING  Use the values   ​-1, 0, 1, and 2​ in the correct box so the graph of each function intersects the x-axis.

f(x)=3x[ ]+1

f(x)= I2x-6I-[ ] (I being an absolute value sign)

f(x)=[ ]x^2+1

f(x)= [ ]

I'm sorry, I hope it doesn't look like I'm just asking for an answer. I genuinely want to know how to solve this as the rest of my homework wasn't too bad. TBF, my teacher did say some of the stuff on the HW we might not know how to do, but I still want to learn how to solve it. Thank you so much for anyone that helps. I have an quiz on Friday and want to be ready. Thanks ^_^

i am doing a mathematical investigation in which i find a piecewise function modeling the shape of an hourglass, use solid of revolution to find the volume and then find a derrivative formula for sandflow through the hourglass over time. I have the piecewise function and both the definite integral and the volume, but i am unsure how to go about finding a diferential equation for sand flow either using granular or fluid model. Any ideas? I have the volume, height, and radius of pinch point as data to use.

I feel like I am close to understanding matrices but not completely. I’m having a hard time thinking about matrices as systems of equations.

Specifically in this post I’m wondering why ax + by decide the x coordinates of the transformed(?) vector? I thought that it was ax and cx that held the information about the transformation of the x-coordinates of the vector

Hi,

Im reading Protter and Jacods probability essentials, and theres one thing i cannot simply understand.
They write:
"Theorem 7.2. A function F is the distribution function of a (unique) prob ability on (R,B) if and only if one has: (i) F is non-decreasing; (ii) F is right continuous; (iii) limx→−∞ F(x)=0and limx→+∞F(x)=1."
But why dont we need left continuity. The borel sigma algebra is symmetric, and thus limits should be preserved not just from the right?

I was just going through the chapter of Probability when an interesting question struck my mind: what is more probable? Randomly shuffling a deck of 52 cards and getting the same exact order or sending a radio wave in a random direction and establishing contact with an alien planet. This had me thinking for quite a long time as both seem equally probable.

To satisfy my questionable taste in humor, I wrote some results of point set topology disguised as crank in an obscure sub. It's ruffled more feathers than I intended, I guess it's called Poe's Law. Is anyone willing to cosign it? I'm pretty confident in it but I'm not infallible.

Familiarity with product spaces, the cocountable topology, and the order topology is expected.

Part 1 Part 2

Im working on a parabolic solar oven for an engineering class, and am struggling with how to find the focal curve of a parabola with rim axes of 14” and 20” that is 4” deep so I know where to mount my receiver, where I will be cooking. I know a standard parabola of revolution focuses to one point but I decided to go with an elliptical paraboloid to maximize the area exposed to light. I cant find any information online and that which I have seems conflicting, with some describing two separate focal points and others describing a caustic curve. Any help is greatly appreciated!

I'm trying to come up with a closed form equation for "N", where N is the integer number of circles of diameter "d" that can fit, in two staggered rows with equal numbers, within a larger circle of diameter "D"?

Note that the small circles d may not (likely not) be tangent, but obviously they need to be close to maximize packing.

Any takers?

Messing around with summation and ran into this.

It's been bugging me for a while and I just want to know "Why doesn't it work like that?" & "How to fix it?/What's an alternative yet equal result?" (Preferably the first but math is hell)

I haven't taken a math class in 6 years and my last class was trig and so I'm retaking it but somewhere else and the way they teach sucks so that's not helping. However, this time it's on me that I'm not understanding it.

The standard form (I wasn't taught this in my previous math class, nor was it explained in this one) is (let's use cosine for example)

y = acos(bx-c) + d

It hasn't taught me + d yet, I'm just on the b part and it's saying to take 2pi and divide by b. All the videos I watch say to do it but don't explain it.