Activity for whybecause‭

Type	On...	Excerpt	Status	Date
Answer	—	A: How can I justify whether I'm able to apply the Gaussian elimination method to this system? I may be misunderstanding the question, but technically you can always use the Gaussian elimination method. But perhaps what you meant to ask is: Will the method give you a unique solution? Sometimes you get no solutions, sometimes you get infinitely many solutions, and sometimes you get precis... (more)	—	about 2 years ago
Answer	—	A: How to write the big Xi notation in MathJax? If you google the Greek alphabet, the LaTeX rendered Xi is a slightly stylized version of the usual print Greek character. This doesn't seem bad to me. Note that in many languages, Greek included, you often produce a symbol in hand-writing somewhat differently than you produce it in typesetting (... (more)	—	about 2 years ago
Answer	—	A: How $ijk=\sqrt{1}$? The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=-1$. How is that possible?! Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another wa... (more)	—	about 2 years ago
Answer	—	A: Is it impossible to prove Jensen's inequality by way of step functions? Right, due to the comment, we know the convex function is continuous and therefore the rest of the proof can be done using the MCT. In particular this is because every function is approached from below by a sequence of step functions. (more)	—	about 2 years ago
Answer	—	A: "Pointwise equicontinuity implies uniform" implies compact Oh wait never mind. The natural number example works because every family of functions is equicontinuous, because delta=1 works for everything. (more)	—	about 2 years ago
Question	—	"Pointwise equicontinuity implies uniform" implies compact Suppose $(M,d)$ is a metric space such that every sequence $fn:M\to \Bbb R$ which is pointwise equicontinuous is also uniformly equicontinuous. Does this imply that $M$ is a compact metric space? My thoughts: If true, we could try to show compactness directly or show sequential compactness ... (more)	—	about 2 years ago
Question	—	Is it impossible to prove Jensen's inequality by way of step functions? Jensen's Inequality: Let $\varphi:\Bbb R\to \Bbb R$ be convex, and $f:[0,1]\to\Bbb R$ be integrable, and suppose $\varphi\circ f$ is integrable over [0,1]. Then $$ \varphi\left(\int{[0,1]} f\right)\le \int{[0,1]}\varphi\circ f $$ A proof from step functions: I have seen a proof of this inequal... (more)	—	about 2 years ago
Answer	—	A: $\{f_n\}\to f$ in $X$ implies $||f||\le\liminf||f_n||$ Oh I think I got it. $$ \lim |T(fn)| = \liminf |T(fn)| \le \liminf ||fn|| $$ (more)	—	about 2 years ago
Question	—	$\{f_n\}\to f$ in $X$ implies $||f||\le\liminf||f_n||$ Let $X$ be a normed linear space and suppose that, for each $f\in X$ there exists a bounded linear functional $ T\in X^ $ such that $T(f)=||f||$ and $||T||=1$. Prove that if $\\{fn\\}\to f$ in $X$ then we have $||f||\le\liminf||fn||$. I think the only $T$ that makes sense to work with is that ... (more)	—	about 2 years ago
Answer	—	A: Missing a solution: Are A and B always equal if A-B=0 Dividing by 5-x is valid only if x is not 5. But x=5 is another solution to the original equation. (more)	—	about 2 years ago
Answer	—	A: Why 1. multiply the number of independent options? 2. add the number of exclusive options? Look at the sub-picture with the beef burger, and all the nodes which extend from it. There are four options. Now look at the sub-picture with the chicken burger. Same number of options. In fact, for each one of those sub-pictures, there are four options, one for each of the different cho... (more)	—	over 2 years ago
Answer	—	A: How's it possible to arrange 0 objects? How can 0! = 1? This explanation may work for some but not for others--it is perhaps a matter of taste or intuition whether you feel that it is sensible. Is doing nothing a "way of arranging" zero objects? Eh, you can get very philosophical about this and probably not in a productive way. I find the following... (more)	—	over 2 years ago
Question	—	Are we in a "history-valley" for Topology? Here is my current rough timeline of Topology: Newton invents the calculus People like Riemann and Cauchy make it rigorous, and by this time, we have the $\varepsilon,\delta$ definition of continuity of a real valued function of a real variable. We realize that we want a definition of the con... (more)	—	over 2 years ago
Answer	—	A: Intuitively, why does $p$ vary inversely with $P(C_3 \mid D_2)$? But directly with $P(C_2 \mid D_3)$? $P(C3|D2)$ is the probability of the car being behind door 3 given that Monty opened door 2. $p$ is the probability that Monty opens door 2, under the assumption that he has a choice between it and door 3. That is to say $p$ is the probability that Monty opens door 2 under the conditions that: (i) ... (more)	—	over 2 years ago
Answer	—	A: Book suggestion category proposal As long as we only answer it once per field, and are good about closing duplicate questions and redirecting them. Because this kind of question gets asked about once per second, somewhere on a math-related discussion forum. (more)	—	over 2 years ago
Answer	—	A: Solely reputable textbooks ought be cited. I agree that canonical textbooks are superior--often because they are in the canon for a reason! But if for no other reason, it's super helpful to have everyone literally on the same page. If you say you learned Calculus and then you mention a bunch of stuff I never heard of, or don't know a bunch o... (more)	—	over 2 years ago
Answer	—	A: How can we grow this community? I think you can moderate the negative impression of not having recent questions, by displaying the number of views that a question gets when you post it. That way when someone posts a questions and sees the views update, they at least know that the post is generating some amount of activity. Th... (more)	—	over 2 years ago
Question	—	$\int_{E_n} |g|^q = \left| \int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g| \right|$ I am trying to understand why the following equation is true. Here $E$ is a measurable set and all functions are defined and measurable on it. $1<p,q,<\infty$ such that $\frac 1 p+\frac 1 q=1$ and $g\in L^q(E)$. $En= \\{ x \in E:|g|\le n \\}$. And there exists a number $M$ such that for every $f\... (more)	—	over 2 years ago
Question	—	Should there be more than one sort of math community? There are, to my mind, two very different but important kinds of mathematical question. There is the kind that the Codidact and SE communities both like: Straight-forward "give the proof/calculation" questions. This can be "How do I solve this?" or "Why is this wrong?" or a few other variations. ... (more)	—	over 2 years ago
Answer	—	A: Without calculations, how can you visualize "that half the squares are white and half are black"? I could argue: Clearly for every white square there is a corresponding black square. This is true row-by-row because the row-length is even and the colors alternate. Therefore it is also true taking all rows together. But I'm not sure why you can't imagine rotating the board. The upper-left w... (more)	—	over 2 years ago
Answer	—	A: Questions with a quote/screenshot and a request to explain I'm pretty averse to using the downvote, so for me this would depend on context. If it's literally just a screenshot with "how?" written after it, yeah, downvote. But in general I think downvoting has more power to harm a community than to help, so I use it with caution. If there's at least a li... (more)	—	over 2 years ago
Answer	—	A: Intuitively, why can $a, b$ cycle in ${\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}$? A possibly helpful further note beyond the answers given elsewhere here: Do you intuitively understand why $ab=c$ is equivalent to $a=c/b$ for all $b\ne 0$? If so, and if you intuitively understand the commutativity of multiplication, then these two intuitions put together give you everything that ... (more)	—	over 2 years ago
Answer	—	A: In general, does $\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$? This is a special instance of the more general fact that $P(A)+P(B)=1$ if $A$ and $B$ "partition" the probability space. I'll explain what I mean. A partition of a set is any way of carving up the set into (1) disjoint and (2) exhaustive subsets. For instance if the set is {1,2,3,4} then a par... (more)	—	over 2 years ago