SQT

> convergent series
convergent sequences? should be the same either way, just curious
> what would the basis of such a space be?
If it's the rationals or reals, the basis is the real numbers.
If it's a general metric space, the Cauchy completion of that space.

Correct. In that case you would have had 1 variable and 2 free parameters. The 2 free parameters give rise to a 2D linear object, which is a plane through the origin.

the basis are not the reals / rationals / completion. the space of sequence is not the same as the space of sequence by the equivalence relation of being co-cauchy.

the basis for the set of convergent series would be pretty complicated. it's an infinite dimensional space. for example in the rationals, all sequences that are eventually constant are convergent, so there you already have an infinite amount of lin ind sequences (think (0, 0, 0, ...., 0, 1, 0, 0, ....), as in, only one 1 and the rest being 0). These are not a basis though, as there are pretty complicated convergent sequences.

axiom of choice guarantees the existence of a basis.
good luck with constructing it though.

the dx/dt term shouldn't be there according to my book. Have I done the differentiation wrong or is there some trick I'm missing?

nvm just realised x is a constant in this context

In pic related, is the author trying to say E ~ F if there is a one-one map from E into F or there is a one-one map from F into E?

he's being intentionally ambiguous: it's either both. a one-to-one map is inversible, so he doesn't want to use asymmetric language

Does one-one correspondence mean bijection?

How do i find an orthogonal complement?

Yes; mind you, one-to-one *function* or *map* is usually meant to be injective. 'Correspondence' is the key word.

Yes. Don't listen to the other guy. One-to-one correspondence gets used all the time; the word doesn't indicate anything in particular.

your retard

Notice the homomorphism f_w:S->k v->
Calculate the kernel K for w=(1,2,1)
Then repeat just this time with K instead of V and w=(2,5,1)
Then you get a 1-dimensional kernel, and it's orthogonal to your whole span S

The orthogonal complement to a subspace [math]W[/math] of an inner product space [math]V[/math] is [eqn]\{ v \in V : \langle v,w \rangle = 0 \text{ for all } w \in W \}.[/eqn]In your case, [math]V = \mathbb{R}^3[/math] and the inner product is probably the dot product, so you need to find a basis for all the vector space of all [math](x,y,z) \in \mathbb{R}^3[/math] such that [math]x + 2y + z = 0[/math] and [math]2x + 5y + z = 0[/math].

This is a system of equations that you can solve in the usual way.

Is there a way to test if the ground in an outlet is working, would this method work?

>charge myself up with static electricity
>measure the voltage between me and the supposed ground
>touch the ground directlywith my other hand
>see if the voltage drops to ~0

Would this work, or would the charge difference equalize the second I start measuring it?

Use the cross product to find a vector v that is perpendicular to both (1,2,1) and (2,5,1). The span of {v} is what you want.

That method won't work.

You can test whether the ground is connected /at all/ using a ground tester, which is basically a high-value resistor in series with a LED. Connecting this between live and ground will light up if the ground is connected.

(You can't just use a mains lamp because you need to a) limit the current to avoid tripping the GFI, and b) avoid electrocuting anyone touching a grounded surface if the primary ground connection is bad).

A better test would use a resistance meter with a long cable to test the resistance between the ground pin and the grounding rod. You also need to measure the resistance of the tester cable and subtract that.

hey OP if you're still there(or anyone who knows). What book are you using that provides answers? I am using Strang's introduction to linear algebra but the lack of feedback on ANY of the questions is driving me insane.

> perpendicular

Nigga, that shit only flies in 2D. Orthogonal is what you're looking for.

being this autistic.

>Nigga, that shit only flies
>>>/worldstar/

I have to prove that 2 graphs, [math]y = x^{2}-4x[/math] and [math]y = 2x-x^{2}[/math] meet when [math]x = 0[/math] and [math]x = 3[/math]

Do i just calculate it like below? I somehow couldn't google the solution to how two parabolas meet

[math]y = x^{2}-4x[/math] where x = 0
[math]y = 0^{2}-4\times0 = 0[/math]
[math]y = x^{2}-4x[/math] where x = 3
[math]y = 3^{2}-4\times3 = -3[/math]

[math]y = 2x-x^{2}[/math] where x = 0
[math]y = 2 \times 0-0^{2} = 0[/math]
[math]y = 2x-x^{2}[/math] where x = 3
[math]y = 2 \times3-3^{2} = -3[/math]

So they both meet at (0, -3)?

Sorry if this question is really stupid but im just getting back into maths after almost 3 years of school (5 years since i did pure maths)

Oh and i only asked because my .pdf textbook answer are just 'self-checking'

compute a Grobner basis for lexicographical order to find solutions. compute Grobner basis in drl order to find the Hilbert Series and thus the dimension, you astronomical faggot.

Since y=x^2 - 4x and y=2x - x^2 :

x^2 - 4x = 2x - x^2 and so 2x^2- 6x = 0 which has solutions 3 and 0.

>That method won't work.
For the reason I mentioned, or because of something else?

>You can test whether the ground is connected /at all/ using a ground tester, which is basically a high-value resistor in series with a LED. Connecting this between live and ground will light up if the ground is connected.
So basically the same as measuring the voltage between live and ground? I guess the voltage stays the same even if the resistance is high though. But does it really matter if the ground wire has high resistance if the ground I'm testing is just going to be used to remove static electricity from myself while working on electronics?

> For the reason I mentioned, or because of something else?
Touching the earth wire will dissipate the charge whether it's connected or not. Also, the meter would need very high impedance to avoid dissipating the charge.

> So basically the same as measuring the voltage between live and ground?
No; testing the ability to conduct current between live and ground.

> But does it really matter if the ground wire has high resistance if the ground I'm testing is just going to be used to remove static electricity from myself while working on electronics?
No.

A safety ground which might need to carry a significant current needs low resistance.

>No; testing the ability to conduct current between live and ground.
But it is conducting current through the multimeter, it's just going through a ~1m ohm resistor. If the voltage measured between live and ground is the same as live and neutral, doesn't that mean that the electricity can (and wants) to go to ground, and proves that the ground works?

Say I have an invertible matrix A with non-negative entries.

Is it true that A*A^T will always be positive definite? (all eigenvalues > 0)

A and A^T will have the same eigenvalues, but in general not the same eigenvectors I think?

What's the difference between [math]P(x;\theta)[/math] and [math]P(x|\theta)[/math]?

I am working out of Bak and Newmann by myself and trying to do each problem that has a solution in the back for feedback purposes, although idk how beneficial that really is. However, I do not understand how they came to this answer and I think what bugs me about it is how they wrote the equation to the plane where it is stated differently in the text prior to Proposition 1.12. I tried to include all the necessary information here.

More specifically I have issue with how they wrote [math]A(x^2+y^2)+Bx+Cy=D[/math] as the equation of the plane. If I can be convinced or visualize that the rest I see is rather easy to get to.

alright, 1: you screwed up your elimination, you could have turned the 3 into 0 and turned the -5 into 1. 2: you can immediately figure out the dimension and the linearly independent vectors comprising the dimension from a correct gauss-jordan elimination. When an elimination is complete, there are some columns of the matrix which lack the "1" entry (which typically forms a diagonal pattern, but if a column can't eliminate in a way that produces the 1 in the correct place, then the diagonal pattern is broken). Make a list of all of the columns that lack the appropriate "1" entry, say the list is 3,6,7,10. Next, calculate what the rows (of your x,y,z...etc. column vector) 1,2,4,5,8 and 9 would need to be for the product of the row reduced matrix and column vector to be all 0s, AND for you to arbitrarially set row 3 as 1, row 6 as 0, row 7 as 0, and row 10 as 0. Write down your column vector. Repeat with row 7 having 1, and the other rows 0 etc. Now you should have written down 4 linearly independent column vectors that are in the kernel of the row reduced matrix. Finally because you can produce the original matrix from a product of inverse elementary row operations, and the row reduced matrix, you know the original matrix has these 4 linearly independent vectors in the kernel, so you have the dimension of the kernel, and some basis vectors spanning the kernel.

Ok I am just fucking stupid.
[math]A(x^2+y^2)+Bx+Cy=D[/math] is just another way to write the equation of a circle. Jesus christ. I guess it's called the stupid questions thread for a reason.

Say I have a matrix [math]A\in\mathbb{R}^{m\times n}[/math] of rank [math]k[/math]. Is there an efficient way of finding [equation]A'=\begin{pmatrix}A_{1,1} & A_{1,2}\\ A_{2,1} & A_{2,2}\endpmatrix[/equation] such that [math]A'[/math] is a permutation of [math]A[/math] and [math]A_{1,1}\in\mathbb{R}^{k\times k}[/math] is regular?

ok seems like i misused the equation tag, that is meant to say
[math]A'=\begin{pmatrix}A_{1,1} & A_{1,2}\\ A_{2,1} & A_{2,2}\endpmatrix[/math]

ok im apparently too stupid, final try:
[math]A'=\begin{pmatrix}A_{1,1} & A_{1,2}\\ A_{2,1} & A_{2,2}\end{pmatrix}[/math]

It says on my multimeter that it has a 10 million ohm input impedance when measuring dc volts. Would that also be true when measuring AC volts? Trying to figure out how much current will flow through the meter when measuring.

For [math]|z| = R(1-2\delta) [/math] why the [math]2\delta[/math].

Can someone show me how to use a determinant of one matrix to find the determinant of another?

I can calculate determinants and the cramer rule and stuff but i'm not sure how to set up the question.

What i have:

[-1 2 1] [matrix1] = [matrix 2]

aei + bfg + cdh - gei - hfa - idb = 2

but i don't know what i'm supposed to do to get the answer of -4.

Determinants are linear in their rows and columns. That means if
[math]A = [\vec{a_1}, \dotsc, \vec{a_k} + \vec{u}, \dotsc, \vec{a_n}][/math]
then
[math]det(A) = det([\vec{a_1}, \dotsc, \vec{a_k}, \dotsc, \vec{a_n}]) + det([\vec{a_1}, \dotsc, \vec{u}, \dotsc, \vec{a_n}])[/math].
And also if
[math]A = [\vec{a_1}, \dotsc, \lambda\vec{a_k}, \dotsc, \vec{a_n}][/math]
then
[math]det(A) = \lambda det([\vec{a_1}, \dotsc, \vec{a_k}, \dotsc, \vec{a_n}])[/math].
Analogous rules hold for rows. Using this, you should be able to figure out your problem. If you're still having trouble, let me know.

Sorry i dont understand the symbol notations :(

When I write
[math]A = [\vec{a_1}, \dotsc, \vec{a_n}][/math]
that means the columns of the matrix [math]A[/math] are composed of the vectors [math]\vec{a_1}, \dotsc, \vec{a_n}[/math].

For example, suppose I have [math]\vec{a_1} = (1,2)[/math] and [math]\vec{a_2} = (3,4)[/math]. Then if I write [math]A = [\vec{a_1}, \vec{a_2}][/math], that means [math]A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}[/math].

why isnt it
1 2
3 4

i thought vectors and equations were written as rows

super confused :(

Is this a correct representation of how electricity is wired in a house?

Neutral and earth (ground) are separate rails. Earth is split from neutral after the meter but before any RCD and connected to a grounding rod.

today i learned about homology and tried to apply it to linear systems
i wanted to see if homology had anything to do with the consistency of a system
word of warning: i learned everything i know from wikipedia

so we have [math] \mathbf{Ax=b} [/math] where A is an m×n matrix, x is a n×1 matrix, and b is a m×1 matrix
we can form a chain complex [math] \mathbf{0\overset{f}{\longrightarrow}\mathbb{R}^n\overset{A}{\longrightarrow}\mathbb{R}^m\overset{g}{\longrightarrow}0} [/math]
i call it the fag complex
where f only maps 0 to 0 and g maps everything to 0
it is indeed a chain complex, g(A(x)) = 0, A(f(x)) = A(0) = 0
so, getting easy stuff out of the way, [math] \text{im}(f)=\ker(f)=\mathbf{0}~,~\text{im}(g)=\mathbf{0}~,~\ker(g)=\mathbb{R}^m [/math]
using the wikipedia definition of nth homology [math] H_n(X):=\ker(\partial_n)/\text{im}(\partial_{n+1}) [/math]
[math] \partial_2=f,~\partial_1=\mathbf{A},~\partial_0 = g [/math] for the boundary functions

so [math]H_1(fag)=\ker(\mathbf{A})/\text{im}(f)=\ker(\mathbf{A})/\mathbf{0}=\ker(\mathbf{A})[/math]
and [math]H_0(fag)=\ker(g)/\text{im}(\mathbf{A})=\mathbb{R}^m/\text{im}(\mathbf{A})[/math]
which is trivial if [math] \dim(\text{im}(\mathbf{A}))=m [/math]
and i know because rouche-capelli, the system is consistent if and only if [math] \dim(\text{im}(\mathbf{A})) = \dim(\text{im}(\mathbf{A|b})) [/math]

i feel that these two things are related, but i haven't been able to make a connection yet
the kind of thing i'm expecting to see is something like "A has a unique solution if H0 is trivial"
how do i move forward from here?

Let [math] \lambda [/math] be an eigenvalue of [math] A [/math]
Since [math] det (A) =det(A^T)[/math] we have [math] 0 = det (A-\lambda I) =det((A-\lambda I)^T) = det(A^T - \lambda I) [/math]
therefore [math] \lambda [/math] is also an eigenvalue for [math] A^T [/math]
Now let [math] x [/math] be an eigenvector for [math] A^TA [/math] with eigenvalue [math] \lambda [/math].
Then we have
[math] \lambda = \lambda \left \langle x,x \right \rangle =
\left \langle \lambda x,x \right \rangle=
\left \langle A^TAx,x \right \rangle =
\left \langle Ax,Ax \right \rangle =
||Ax||^2 \geq 0[/math]
We also know, that [math] A [/math] is invertible and therefore [math] A^T [/math] is too (can't have [math] 0 [/math] as an eigenvalue because [math] A [/math] doesn't have it).
From that we can conclude, that [math] A^TA [/math] is also invertible, which means [math] \lambda > 0 [/math]

oh and I didn't write, that I assume [math] ||x|| = 1 [/math] w.l.o.g.

Anyone know where I could download a PDF copy of Physics in Minutes by Giles Sparrow? I've looked everywhere but can't seem to find it...

(Co-)Homology can be defined for objects of any abelian category.

Vect_k, for any field k, is abelian. So you don't need to do anything special to develop (Co-)Homology. Just follow the usual procedure.

What activity in this planet requires knowledge of this seemingly useless material?

In older installations (at least in my country) the ground and neutral is combined into something called PEN-leader (protective-earth neutral), and there is only 1 rail.

That seems a bit weird though, because if I were to touch ground in an outlet, then connect myself to another self made connection to earth, wouldn't some of the power going back to neutral, travel back through me from the rail, or will it always pick the best way? Same question could be asked with your seperate system, when there is a fault and a device leads current to the ground. (assuming you don't have a RCD).

> if I were to touch ground in an outlet, then connect myself to another self made connection to earth, wouldn't some of the power going back to neutral, travel back through me from the rail, or will it always pick the best way?
Current divides in inverse proportion to the resistance. So if the resistance of your body is 10,000x that of the neutral wire, it will carry 1/10,000th of the current.

> Same question could be asked with your seperate system, when there is a fault and a device leads current to the ground. (assuming you don't have a RCD).
A short between live and earth (e.g. the casing of the faulty appliance) will blow a fuse or trip a breaker. A short between neutral and earth will trip the RCD (these have been mandatory for some time), as will a low-current leak between live and earth.

The difference between neutral and earth is that earth doesn't carry any current in the absence of a fault.

If you were to just connect metal surfaces to neutral, then touching two such surfaces, where one of them is much farther than the other from the panel, could result in your body conducting a non-trivial amount of current. Even a milliamp across the chest (i.e. arm to arm) can cause fibrillation, and that's plausible if you're bypassing a stretch of neutral wire which is relatively long and/or has less than perfect connections.

i didn't know that and that's cool but that's not what i was going for
i was trying to establish a connection between homology and the solvability of a linear system

i don't even know, i'm just a hobbyist

>A short between neutral and earth will trip the RCD
How would it even detect if earth and neutral were connected? The voltage between them is 0v. Or do you only mean in an event where a faulty device would send equal current through ground and neutral, current that normally wouldn't trip it if it were coming only from neutral or ground?

Ground and neutral is connected at some point, so what's stopping the power going out through neutral from going back through ground, out the connection in the outlet through me, and down into my earth connection?

There are elementary operations that you can perform on the matrix you were given such that you have a term that is the determinant of A. If you keep track of the ways in which the elementary operations change the determinant, you can calculate the determinant you are trying to find.

> How would it even detect if earth and neutral were connected?
An RCD detects an imbalance between live current and neutral current. If the live current returns by any path other than the neutral wire, the two won't be equal and (if the difference is enough), the RCD will trip.

This will happen if the live current returns via the earth wire, or via a direct connection to earth (e.g. dropping a power tool in your garden pond), or via your neighbour's earth wire (e.g. they cut through your garden lighting cable with a hedge trimmer, connecting your live to their earth).

> Ground and neutral is connected at some point, so what's stopping the power going out through neutral from going back through ground, out the connection in the outlet through me, and down into my earth connection?

Mostly the fact that the point at which neutral and earth are connected is the same point that the wire to the earthing rod is connected, so the "long route" through you will be much worse (and any imperfections in the house wiring will reduce the current rather than increasing it).

cleaner way is to use definition of
A is positive definite iff x^T A x > 0 for all non zero x.

Then x^T A^T A x = |Ax|^2 > 0 for any injective A (this also works for non-square matrices)

how do i find the maximum vertical distance between 2 parabolas ([math]x^{2}-4x[/math] and [math]2x-x^{2}[/math] when the vertical separation between the graphs is [math]6x-2x^{2}[/math] for [math]0\leq x\leq 3[/math]?

there isnt meant to be a '(' before the first parabola

>I don't want to turn Veeky Forums into my personal homework help, but I've been getting confused with the statistics for my dissertation and my supervisor is a fucking idiot qualitative researcher who just shrugs and ignores me whenever I ask him for help, so wondering if anyone could help me out

I used a questionnaire to measure a certain personality construct (X), which can either be conceptualised as a single higher-order construct or five separate facets (X1, X2... X5). My results show low intercorrelations between these facets, so I think it's best to go with them rather than the higher-order construct. Scores for each of these are on a continuous scale between 1 and 7.

I want to see if any of these are associated with performance at two timepoints (Y1, Y2), also measured on a continuous scale. Furthermore, I've also measured the use of two techniques (Z1, Z2), also on a continuous scale. Oh, and just in case it wasn't complicated enough already, I also have a covariate (C) which is likely to have the greatest impact on Y1 and Y2.

Basically, what I want to find out is (a) whether X1, X2, X3, X4 and/or X5 are associated with high performance at Y1 and Y2 (after controlling for C), and (b) whether Z1 and/or Z2 mediate this effect.

(1/2)

However, my data isn't being too kind to me and I'm getting fucking confused trying to work it out. For the first hypothesis, I tried two multiple linear regression models with Y1 or Y2 as the outcome variable, and then having C as my first predictor 'block', then X1, X2, X3, X4 and X5 as additional predictors in the second block. Both times, the ANOVA for the overall model, but none of the individual predictors were significant. Anyone know why this has happened and what I should do next? Should I report it as an insignificant finding, or is it weak but significant? Or have I likely made some mistake somewhere that I need to correct?

I thought maybe I was making it too complicated, so just loaded all of my variables into a correlation matrix to try and understand the data better. This made a bit more sense, and I found some of the expected results (e.g. C was strongly correlated with Y1 and Y2 - Z1 and Z2 were correlated with Y1, Z1 only was correlated with Y2 - X4 and X5 were correlated with Y1, X3 and X5 were correlated with Y2). I now have a general idea of what's going on, but is this enough in terms of statistical analysis? Would I need to do any further tests in order to return to my original hypotheses?

Thanks in advance for any help on this, because I'm really confused and the staff at my university are fucking useless and refuse to help. I'm convinced that there is some easy method to do this, but it goes beyond the scope of what I learned in my stats modules, and at the moment I'm stuck.

(2/2)

> how do i find the maximum vertical distance between 2 parabolas
> x^2-4x
> and
> 2x-x^2
For any given x, the vertical distance is their difference, i.e.
2(x^2)-6x

This has roots at x=0 and x=3, thus its peak is at x=3/2, when its value is -9/2.

...

Could anybody give me general study tips and advice? Specifically, I need to study and refresh myself on a lot of material before starting grad school next month.

For Statistics, I'm not sure how to approach it since I'm awful at maths and I would imagine studying maths is entirely different than anything not-maths or physics (since no equations/problems, that sort of thing.)

Otherwise, for strictly information memorization, I was going to read a certain amount every day, try to pay attention to specific important information, make flash cards to help with memorizing and retaining key terms, and I'm not sure where to go beyond there.

don't focus on memorizing. learn the material and you'll remember it.

a tip would be to approach the material in a way that you are specificially not trying to memorize it.

a math analogy would be learning how to derive a formula from something basic rather than memorizing the formula

that's how you should approach all learning. if you specifically avoid trying to memorize, you'll learn a lot more and remember it well automatically.

as for simple "facts," to remember them just inject as much emotion into them as possible. our memory works by sensory perception. the more senses you involve, the stronger the memory will be.

make it funny/sad, colorful, physical, invent a story, invent something out of the ordinary for it, etc.

Well I have memory problems which is why I mention memorization as that seems to be one of the biggest problems I have when it comes to material. As far as math, I would have to memorize equations because I wouldn't simply understand them unless I had a personal tutor helping me along with that.

Come on Veeky Forums you're meant to be full of autistic stats geniuses. I'm just a lowly psychology student and that's not even a real science, so this shit should be easy.

What does locality mean in category theory?

bump

Why can't I measure the resistance of an LED? It lights up if I have continuity mode on, but not when measuring resistance.

> Why can't I measure the resistance of an LED?
Because the relationship between current and voltage is highly non-linear, whereas "resistance" implies that voltage is (roughly) proportional to current.

> It lights up if I have continuity mode on, but not when measuring resistance.
Continuity mode probably uses a higher voltage. You need around 2V to get a red LED to conduct.

Pretty sure you fucked up RRE since the rank of the original matrix is 2 and the rank of your RRE matrix is 1

How much is the current usually limited to when using continuity mode/resistance measuring? Is there a way to check, without another meter? I got the voltages from charging a cap with it, and the regular resistance mode was 0.2v while continuity was 2.2v.

I don't get how powering the LED with 1v + 50ohm resistor, is any different from powering it with 5v + 250ohm resistor. Both will give it around 20mA. I know it has something to do with the voltage drop, but I don't understand how that can be when logically, the "strength" of the power should always be V/R.

1V won't even turn it on; you need around 2V for a LED.

Below the threshold voltage, a diode conducts almost no current. Above it, the current rises very quickly.

With 3V, you'd need 50 ohms for 20mA: 2V across the LED, 1V across the resistor, divided by 50 ohms = 20mA.

With 5V, the drop across the resistor is 3V, so you'd need 150 ohms for 20 mA.

If you select a resistance range manually, the current will be whatever gives 200mV at the top of the range (200mV is the typical sensitivity of the DAC+amplifier, without any gain or attenuation).

E.g. 0-20kOhm will use 10uA so 20kOhm=200mV. If it's auto-ranging, it will select whichever range keeps the voltage between 20mV and 200mV.

Continuity probably uses a fixed voltage with a current-limiting resistor. A higher voltage will allow diodes (e.g. a rectifier or polarity-protection diodes) to conduct.

How do I become a top student?

Going into a physics undergrad program soon.

I'm in my senior year of my physics degree and I decided to get a minor in EE.

How time consuming are EE classes?

I'm taking
>Modern Digital Systems
>Electronics
>Microcontrollers
>Math Methods for Physcicsts
>English

Plus I'm TAing, doing research, and applying to grad school. Is this too much for a single semester? Should I drop something?

A family of sets [math]\{A_i\}[/math] is a function [math]A: I \to X[/math], where [math]A_i := A(i)[/math].

The cartesian product over a family is the set of functions from [math]a: I \to \bigcup_{i \in I} A_i[/math] satisfying the property [math]a(i) \in A_i[/math] for each [math]i \in I[/math]. Each function [math]a[/math] is of course a family indexed by [math]I[/math].

Axiom of choice states that the cartesian product over a non-empty family of non-empty sets is non-empty.

My question is, why do we need an axiom to assert this? Suppose [math]\{A_i\}[/math] is a non-empty family of non-empty sets. Then we define [math]a: I \to \bigcup_{i \in I} A_i[/math] by setting [math]a_i[/math] to be some element of [math]A_i[/math], which must exist, because each [math]A_i[/math] is non-empty. Then [math]a[/math] would be in the cartesian product, so it cannot be empty.

Clearly there's a problem with my above construction, but I'm not sure what the problem actually is.

Say you've formally proven

[math]\forall (i\in I) \, \exists a.\, a\in A_i[/math]

Let P(f) be the formula expressing f is a function (e.g. modeled as a pair of... blabla). You need to prove

[math]\exists f.\, P(f) \land \forall (i\in I) \, f(i)\in A_i[/math]

Well you can't from the other axioms of ZFC. It has strong comprehension schemta but they don't suffice.

Yes, you're right that if e.g. all [math]A_i[/math] are well ordered, you could use "inf A_i" and this would denote an object, and you could use it (via comprehension) to express an f. But from
[math]\exists a.\, Q(a)[/math]
you can't actually refer to any [math]a[/math].

The ordering thingy is the reason why e.g. the claim that all sets can be well-ordered is equivalent to choice.

Choice is btw. provable in some constructive theories liky Martin-Löfs Type theory, because there, in that logic, proving existence always means to construct something, and then you may just use those constructed a's to set up an f.

>Clearly there's a problem with my above construction
well you have to CHOOSE infinitely (maybe > aleph0) many elements for this to work, so I assume thats where we need that axiom of CHOICE

Having a lot of trouble with number 8. I can't use trigonometry. I tried using the midpoint theorem, but it didn't amount to much.

Can someone point me in the right direction?

Depends on the school.
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trigonometry

Add a line from C to the midpoint of AB. You've now split the triangle into equal sixths. a2 and a4 each comprise two sixths.

If you don't see why the six triangles are equal, start with an equilateral triangle, where they must be equal by symmetry, and note that shearing doesn't change areas and scaling scales all areas by the same factor leaving ratios unchanged.

Does [math]P\big( (A \cap B')' \big) = P(A' \cup B)[/math] ??

I'm going to assume P denotes the power set and ' denotes the complement. No. Note [math](A \cap B') =B \setminus A[/math], which is not necessarily equally to [math]A' \cup B[/math]. The first set corresponds to things in B AND not in A, while the second set is things in B OR not in A.

From this though, it should be clear the former set is in fact a subset of the latter set.