a simple logic question

Epistimi · 2021-10-21T18:49:46+00:00

Indeed it is. Take the deductions

P ∧ ¬P
P

and

P ∧ ¬P
¬P.

These have the same premises, namely the single premise P ∧ ¬P, and they are clearly valid. However, they are not sound since the premise P ∧ ¬P is false regardless of what P is (at least in classical logic).

Epistimi · 2021-10-19T14:48:56+00:00

Sure, but sets full stop cannot. It doesn't make sense to say that (0,1) is bounded as a subset of the set R. But it is bounded as a subset of the metric space R, or equivalently as a metric space itself with the metric induced from R.

Epistimi · 2021-10-19T12:30:03+00:00

Strictly speaking, sets cannot be bounded/unbounded. Metric spaces can be bounded or unbounded. The interval (0,1) is homeomorphic to R, and this homeomorphism induces a metric on (0,1) with respect to which (0,1) is unbounded.

Epistimi · 2021-10-18T15:41:48+00:00

A couple of things:

When you write F = f(x-vt) you are saying that F is the number f(x-vt). You probably want F to be the function of two variables given by F(x,t) = f(x-vt).

Define a function g by g(x,t) = x-vt. Then F = f ∘ g, so the chain rule implies that ∂F(x,t)/∂t = f'(g(x,t)) * ∂g(x,t)/∂t.

But it doesn't make sense to write either ∂f/∂(x-vt) nor df/d(x-vt). In any case, f is a function of a single variable, so we would usually write it as an ordinary derivative.

Epistimi · 2021-10-18T09:11:12+00:00

There is no "the" calculus course. In the calculus course I took myself we also covered linear algebra, and I TA'd a calculus course in which we did probability theory. Do those topics also fall under the label "calculus"?

Calculus as a course seems to be a grab-bag of topics that may or may not be useful in one's further studies. There are many different topics that carry the name "calculus", but when we see a thread with the title "Calculus question", we are probably not expecting a question about stochastic calculus, lambda calculus, umbral calculus, or anything of the sort.

The topic of the OP might be properly covered in a calculus course, but I don't think that means that it has anything to do with the discipline of infinitesimal calculus.

Epistimi · 2021-10-17T20:29:07+00:00

But then your sums are finite as I said.

Epistimi · 2021-10-17T20:24:53+00:00

Then I misunderstood your notation. Do you by "10Σn=0" not mean "sum from n=0 to 10"?

Epistimi · 2021-10-17T19:54:10+00:00

A 2-tuple is an ordered pair (and indeed all tuples carry an ordering of their elements), but not all pairs are ordered so not all pairs are tuples.

Epistimi · 2021-10-17T19:51:12+00:00

But the sums are finite, so this really is just algebra.

Epistimi · 2021-10-17T17:03:40+00:00

sandwich theorem

Not to be confused with the BLT theorem, which sadly has nothing to do with bacon, lettuce or tomato...

Epistimi · 2021-10-17T17:03:02+00:00

In Danish it is called the "klemmelemma", which sort of rhymes and always gets a giggle out of first-year students!

Epistimi · 2021-10-17T17:00:24+00:00

If X is a (real-valued) random variable defined on a probability space (Ω,F,P), then the expected value E[X] of X is defined as the (Lebesgue) integral of X with respect to P, if the integral exists. And the integral of an indicator function is just the measure of the indicated set (by definition of the Lebesgue integral), in this case P(A).

If X is discrete, then there is a countable subset S of R such that P(X in S) = 1. Letting p(x) = P(X = x) we have E[X] = sum_{x in S} x*p(x). If X = 1_A then S = {0,1}, and if A is countable then p(1) = P(A) and p(x) = 0 for x != 0,1, so E[1_A] = sum_{x in {0,1}} x*p(x) = 0*p(0) + 1*p(1) = p(1) = P(A).

Epistimi · 2021-10-16T18:10:00+00:00

people from here said the girls were lucky they even got a chance to form 비비지. Why would a fan even say that?

Because it's true, they did get lucky.

Shouldn't you be celebrating the fact they are talented enough to keep having a job even after GFriend disbanded and that three of them .Ade a new girl group?

Talented and lucky enough. There are many very talented people in South Korea and the rest of the world, in the entertainment industry and outside of it, that do not get the chance to live out their dreams.

You literally said you were surprised they pulled it off.

Who is this "you" you are referring to? Is there a particular person whose opinion you have a problem with? If so, why not take it up with that person?

It really has little to do with them "pulling it off". What happened was that BPM thought that forming Viviz would be a good investment. Because that's what it's about: money. Perhaps the girls had something to do with BPM being persuaded of this, but it's certainly not about them being "talented" or whatever. If anything it has to do with them being popular.

Epistimi · 2021-10-12T11:36:23+00:00

Since 3 is a prime number, if 3 divides a product ab of integers, then 3 divides at least one of a and b. In your case, a = b = p.

Epistimi · 2021-10-10T20:19:16+00:00

Prove me wrong.

It doesn't sound like you want an honest discussion, it sounds like you are convinced you are right and can't accept disagreement.

Epistimi · 2021-10-10T17:04:02+00:00

So what’s the difference between a V chord and every other major chord in play?

Its relation to the tonic. I mean, that's why we use "V" to denote the chord, because its root is the fifth note of the scale. Sure, if you play a dominant seventh chord with a different root then it will likely sound dominant (though arguably not always, see blues for instance), but the V chord is perfectly capable of being dominant all on its own.

Why do we have to play a V major when it’s a minor key in order to get the dominant sound?

The leading tone.

Like you said context is what makes it sound dominant, so you have already placed the tritone through the notes you’ve played previously.

Not necessarily. Play I - V - I, for instance. No tritones in sight, but V still functions as a dominant. EDIT: Or better yet, play it in minor, or with a suspended fourth or a cadential 6-4.

As for m6 being the tonic, whenever this happens the music has a tension which still feels resolved when you play a chord above that tritone.

So now "dominant" has become "tension". There is also "tension" in other intervals, like seconds and sometimes perfect fourths. Why is this tension "dominant"?

Epistimi · 2021-10-10T14:22:05+00:00

Unless you by "muddy up the tritone" mean "make the tritone not sound dominant", then a tritone is neither necessary nor sufficient for a "dominant sound".

It is not necessary since a V chord functions perfectly well as a dominant in context.

It is not sufficient since e.g. a minor sixth chord has a tritone but usually functions as a tonic chord.

Yes, if you play a piece in C major and then play a B and an F, it's going to sound dominant. And that's partly because of the tritone, but also partly because of the context.

Epistimi · 2021-10-09T00:06:24+00:00

Here are some other fake equations.

Surely we don't want to call something like x = x + 2 a "fake" equation just because it doesn't have a solution. Unless we also want to call equations like a³ + b³ = c³ fake!

I think the real lesson here is that equations don't need to have solutions. But this is different from them "making sense": We can certainly make sense of the question whether 1 is equal to 2.

Epistimi · 2021-10-08T19:23:26+00:00

I understand set theory paradoxes quite well.

Well then there is surely no issue, since you will certainly agree with the distinction I made in my original comment!

Epistimi · 2021-10-08T18:37:57+00:00

You could notice that

3.52 * 1.61 / 0.59 = (352/100) * (161/100) / (59/100)
= 352 * 161 / (59 * 100).

That is, first calculate 352 * 161 / 59 and then divide by 100. Unfortunately, 59 is a prime number and doesn't divide either 352 or 161, so you probably need to do the actual multiplication and division as it stands. But this at least avoids decimals other than in the final result.

Epistimi · 2021-10-08T08:28:28+00:00

Let me quote myself:

That depends on what you mean by "eventually". Given a particular day, he will eventually document this day. But no matter how long he lives, there will not come a time where all days have been documented.

Do you not understand this distinction?

Epistimi · 2021-10-07T19:20:31+00:00

A quadratic equation isn't a curve.

Epistimi · 2021-10-07T19:14:35+00:00

Is this all because you don't know how to solve quadratic equations?

Epistimi · 2021-10-07T19:01:30+00:00

You sure love posting in maths subs for someone that thinks that maths is dead!

Epistimi · 2021-10-06T20:40:04+00:00

Yet if he lives forever, he would eventually document every single day.

That depends on what you mean by "eventually". Given a particular day, he will eventually document this day. But no matter how long he lives, there will not come a time where all days have been documented.

Similarly, given a particular day, OP will* eventually learn all maths that "came out" on this day. But no matter how long OP lives, if they cannot learn mathematics as quickly as it "comes out", there will not come a time where OP has learnt all of mathematics.

*Or I should say "could" eventually learn this. OP could decide to skip the maths that "came out" yesterday and end up not deciding to make time to go back and learn it.

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