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Some authors use different notation, i.e., for $f\colon A\to B$, the associated direct image function is denoted $f_\to\colon \mathcal P(A)\to \mathcal P(B)$. However, since this association is very common it is standrd to simply write $f$ for the extended set valued function, without warning as you say. Some care must be exercised, since for instance if $... 3 The definition of$f$isn't changing.$f(U)$is simply standard notation for the image of$U$under$f$, and similarly for the inverse image$f^{-1}(U)$. Of course$f:X \to Y$is not the same function as its direct image function, which you could denote$\mathcal{P}(f):\mathcal{P}(X) \to \mathcal{P}(Y)$if you really wanted. If category theory interests you,... 4 It's overloaded notation, for sure. But I have never seen it mean anything other than the following: Let$f : X \to Y$. Then, for$A \subseteq X$,$f(A) := \{ f(x) : x \in A \}$. The reason there isn't much notice is probably because it's a very common convention. It may even be explicit at some point in some books on Real Analysis. It turns out that ... 1 Although OP narrowed down the post, there are still many more important historical facts which should be addressed to adequately answer the question, than I can give in this answer. Nevertheless here are some aspects, which might be interesting. At least we will see, OP is right when he thinks that many different candidate definitions of manifolds competed ... 0 Unfortunately I discovered that I've mixed the notation in a work on Lie Groups. It was unfortunated because I than realized that I could use$exp(X)$for the application between the Lie Algebra and the Lie Group, while using$e^X$for the proper matrix exponential and then I could conclude in a theorem that$exp(X)=e^X$. It would have been a great theorem.... 1 As a child, I learned logic with the book "The Game of Logic", by Lewis Carroll. This does not use the modern symbols, but I think that it is a great book to introduce the basic concepts. 1 The whole point of "indexed family" is that it is a function. It is a function mapping the index to some object. So if you consider the first statement, it just says in a more complicated language that every family of non-empty sets has a choice function. Why more complicated? Because it requires the understanding that an indexed family is a function, ... 5 The mostly commonly used AC has 3 forms: choice function: suppose you have a bag of inhabited sets$A$, then you can just say: oh, let$f$be a function on$A$, with the action$f(x)\in x$for each$x\in A$. zorn's lemma: suppose you have a partial order$A$with the property that every linearly ordered subset (chain) is bounded above, then you can say: ... 1 A possible sub-base would be to take balls of only rational radius. Any ball of irrational radius can be realized as the infinite union of balls of rational radius. E.g.$ \ \ B_\pi(x) = B_3(x) \cup B_{3.1}(x) \cup B_{3.14}(x) \cup \cdots$I hope I've interpreted the definitions correctly. I'm taking "sub-base" to mean a subset$S \subset B$that serves ... 1 If$R$is a finite commutative ring with maximal ideals$I_1,\dots,I_k$then by the Chinese Remainder Theorem there is a surjective homomorphism $$R\to R/I_1\times\dots\times R/I_k,$$ and so$\vert R\vert\geq\prod_{i=1}^k\vert R/I_i\vert$. Since each$R/I_i$has at least two elements, this gives$\log_2(\vert R\vert)$as an upper bound for the number of ... 1 @Peter Smith wrote: So it is worth noting that e.g. John Corcoran can write "Three Logical Theories" as late as 1969 (Philosophy of Science, Vol. 36, No. 2 (Jun., 1969), pp. 153-177), finding it still novel and necessary to stress the distinctions between different types of logical theory. Here it is https://www.academia.edu/9855795/Three_logical_theories 0 Overall I agree with Jack D'Aurizio's comment: the best way to snap out of this is to realize and internalize the fact that mathematics is not about "doing well," it's really about collaboration and discovery. It's about talking to people, some of whom you might consider to be absolute geniuses, asking questions, getting stuck and then asking more questions. ... 0 This is along the lines of a nonmeasurable set, and Feynman may have rejected it on physical grounds, but theorem: There exists a nowhere dense set with positive measure, a fat cantor set 2 Well because you only have exams a few times a semester I can see where the perfectionism might come in. A couple of computational errors and you already are down to an A- (or worse!) depending on the grader. Also if the courses are not challenging you might feel like you are obligated to get a perfect score because of your aptitude. Also because math is ... 1 Even if you can obtain a "100%" grade, don't be complacent about it: a stricter examiner might have given you only 98%! You can still be a perfectionist; but what you are presently trying to perfect is the wrong thing. You are trying to perfect the output, whereas what you should be trying to perfect is something like the ratio of output to input (according ... 4 Let me add to Lee Mosher's remarks about math and CS: First, I've done both, and made plenty of mistakes in both places. But in CS, because there are ways of testing things relatively easily, the mistakes are often easier to find. A consequence of this is that you end up developing habits that keep you from making mistakes in the first place. Here's a (... 7 A great mathematician once said to me : making mistakes is not a big deal. What's important is to make the right mistakes. You can interpret this in whatever way you want, but I think the point is this : getting a 96% because you computed something fast in Calculus and disregarded a + or - sign is not relevant. You will still get A+ and there is a reason ... 6 In the bigger scheme of things (beyond grades assigned in undergraduate courses), mathematics does not self-select for perfectionism, nor does it award perfectionism to any realistic extent. Yes, you need to be able to calculate accurately. However, there's a lot more to mathematics than calculation. To be an accomplished mathematical professional, you have ... 3 I'm a perfectionist, but I unintentionally make lots of careless mistakes all the time. Like you, I feel a sharp twinge of disappointment or embarrassment, depending on the situation, but it goes away almost immediately. The reason is that I consider intention and not outcome to be the deciding factor, and so as long as I had put in what I deem sufficient ... 0 To build up statistics from first principles requires measure theory as probability theory is just measure theory on a space defined to have a measure of one, (i.e. a measure space$(\Omega, \mathcal{F}, \mathbb{P})$, where$\mathbb{P}(\Omega)=1$). Then a random variable is a function$X:\Omega \to E$where$E$is another measure space, where we usually ... -4 I might get scolded for this, but statistics to me does not have that much calculus involved in it. However, one big calculus topic is the normal distribution which turns out to be just a really difficult function to integrate which is done so by using the Gaussian Integral which can be proved with the Jacobian Conjecture. Standard deviation and variance are ... 0 I remember reading the abstract of an article (or description of a book perhaps) that claimed to answer this using the principles of evolutionary biology; essentially, the author performed various simulations suggesting that organisms that take, as their fundamental logic, anything other than$2$-valued boolean logic tend to die off in the long run. I think ... 2 Not sure if it answers your question, but you can define them as follows:$\color\red{\neg{x}}$as$\color\green{2-x}\color\red{{x}\vee{y}}$as$\color\green{({x}+{y})\bmod3}\color\red{{x}\wedge{y}}$as$\color\green{({x}\times{y})\bmod3}\color\red{{x}\implies{y}}$as$\color\green{\neg{x}\vee{y}}\color\red{{x}\oplus{y}}$as$\color\green{[\neg{...

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As mentioned by skyking, your definition does not work. Take any statement $P$ that is not provable (Your example of "$x \in S$" is one if you allow non-sentences). Then $( P \lor \neg P )$ is unconditionally true, but you would have assigned it the 'truth-value' $2$. So your definition is self contradictory. However, what you seem to want is Kleene's 3-...

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We don't know that know that the logic we are using works. There are an uncountable number of possible logical systems and there can't be a logical reason to select one logical system over another. The decision to select a logical system is more the result of history and capacity to create results that are useful and pleasing to mathematicians.

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Mathematics is about ideas (sometimes really great ideas) which must be formulated, because there are no visible mathematical objects in reality to rely on. We formulate these ideas using some human language enriched with mathematical symbols and definitions (also possible to express in human languages). The primary mathematical definition is 'mathematical ...

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Coming back to this question a few years later, I've found an example of a game out there that does exactly what you intend your "wager system" to do, and it's even specifically designed for true-false questions. It's called "the credence calibration game", which is a game where you place confidences on how confident you are of a particular answer, from 50% ...

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GÃ¶dels incompleteness theorems are widely interpreted to mean that it is impossible to construct a reasoning system that can prove itself to be consistent. As such, it is almost universally believed that it is impossible to construct mathemathics without starting with some "self evident" axioms. The Wikipedia article explains the details of the theorems ...

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Claim: There is an algebraic procedure that terminates in finite steps to obtain the eigenvalues of a general $n\times n$ matrix. This is true for $n \leq 4$ but fails for $n \geq 5$, thanks to Abel-Ruffini.

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Answer to part 1: If $f$ is an isomorphism of algebraic varieties, it must be a regular map (which is therefore analytic, as polynomials and their appropriately-taken quotients are analytic). Similarly, it's two-sided inverse must also be regular and thus analytic, and so we have that $f$ is also an isomorphism of analytic varieties. Therefore it must be a ...

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The Cartesian product will work, but you want to specify how the concatenated vector is composed. $$\left\{\vec z\in X{\times}Y ~\middle\vert~ \vec z=\vec x\Vert\vec y\,, \vec x\in X\,, \vec y\in Y\,,\mathbf 1^\top\vec x=1\,, \mathbf 1^\top\vec y=0\right\}$$ Or more elegantly: $$\left\{\vec x \Vert\vec y~\middle\vert~\vec x\in X\,, \vec y\in Y\,:\,\mathbf ... 0 Why not? Because X\times Y=\{(x,y): x\in X,\, y\in Y\} so no matter what set is X or what set is Y. 2 The MacTutor biography mentions "Disquisitiones generales circa seriem infinitam" as introducing the hypergeometric functions. This was not his doctoral thesis: it is dated 1812. His doctoral dissertation "Demonstratio Nova Theorematis Omnem Fvnctionem Algebraicam Rationalem Integram Vnus Variabilis In Factores Reales Primi Vel Secvndi Gradvs Resolvi Posse"... 5 Here's a good translation. Gauss' original writing is a little old fashioned but still perfectly readable. http://archive.larouchepac.com/node/12482 The flaw doesn't have anything to do with hypergeometric functions as far as I know. The trouble is he assumes without proof that a real algebraic curve that enters a circle must leave it again. 1 Recently, I come up with the following conjecture, which is proved to be true for small matrices and false for large matrices. Conjecture. Let E \in \mathbb{R}^{d\times d} be the matrix of ones and M \in \mathbb{R}^{d\times d} be a bounded matrix representing a metric. That is, M satisfies the following conditions: 1 \geq M_{ij} \geq 0\  ... 0 In order to sum up the above comments, the corresponding books for these nick names are 1. Baby = Principles of Mathematical Analysis; 2. Papa/Big = Real and Complex Analysis; 3. Grandpa = Functional Analysis; and it seems that the difficulty of contents of the books grow with the age of the nick names! Firstly, you are a baby and things are easy ... 0 It is not merely a restatement. For instance, note that functions can have the intermediate value property without being continuous! See https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)#Darboux_function Also it is not enough for there to be an f(c) it must be between f(a) and f(b). Moreover we are not just saying there is sooome value we are ... 3 According to Earliest Known Uses of Some of the Words of Mathematics, the term "characteristic function" in this sense (actually its French equivalent "fonction caractÃ©ristique") was first used by PoincarÃ© in 1912 (except that with his notation, that function was what we now call the moment generating function). To me as an analyst, the terminology never ... 4 If X is non-empty then we can choose some x_0\in X and define f(x)=x_0 for all x\in X. This is a contraction with a fixed point. 2 Good question! (+1 for the same) Wikipedia is correct here. And it will be best to explain this using an actual concrete example. Let f be differentiable at a and let f'(a) > 0. Using limit definition of derivative and taking \epsilon = f'(a)/2 we see that there is a \delta > 0 such that$$\left|\frac{f(a + h) - f(a)}{h} - f'(a)\right| <...

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A soliton (or, more precisely, a ''one-soliton'' solution) usually refers to a solitary wave solution to an integrable equation. The one-soliton solution together with the multi-soliton solutions of an integrable equation can usually be obtained through the technique of inverse scattering (which requires a Lax pair). So, in a nutshell, we call a solitary ...

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There is no universal definition for "well defined." Here are three common uses of the term. $X, Y$ are sets, $\sim$ is an equivalence relation on $X$, $\tilde{X}$ is the set of subsets $S$ of $X$ having the property that $x, y \in S$ if and only if $x \sim y$. A typical element $S$ of $\tilde{X}$ is often written as $\overline{x}$, for some $x \... -1 The problem has been solved since 1993 with the advent of turbo codes. Tribute to Mr Berrou and Glavieux who discovered those iterative decoders and the concepts of exchanging extrinsic information. Then McKay (unfortunately he just passed away) rediscovered the LDPC codes (Gallager codes). Today the research is focused mainly in which system has the lowest ... 3 I'm a third year graduate student who had to pass three qualifying exams. I used the same strategy to study for all three. This is what worked for me. Step one (general comprehension): Spend two to three hours a day reading proofs and important results for the exam you're studying for. Try to work examples as well as understand theorems. Memorize the ... 2 Jean Goubeault-Larrecq has written a detailed textbook-level introduction to "Lawvere" metric spaces from a topological point of view in Chapter 6 of his book Non-Hausdorff Topology and Domain Theory. He defines a hemi-metric space to be a set$X$equipped with a set-function$d\colon X\times X\to[0,\infty]$such that$d(x,x)=0d(x,y)\leq d(x,z)+d(z,y)$... 4 This is another example of "equivalence instead of equality". Monic maps "ought" not be characterized by$f(x) = f(y) \implies x=y$: the right characterization ought to be$f(x) \cong f(y) \implies x \cong y$. (note: the following was written around the symmetric version; where we further assume$d(x,y) = d(y,x)$. Without this assumption the details are ... 5 I assume you are looking for some real world examples of pseudo-metric spaces (that correspond to what you call Lawvere-metric spaces). Well a family of examples is given by the$\mathcal L^p$-spaces (where$p$is real number in$[1,\infty]$). In general for$(\Omega,\mathcal F, \mu)$a measured space, we have the vector space$\mathcal L^p(\Omega)$whose ... 2 The standard way to write this is$f(x)^2$(this always means$(f(x))^2$rather than$f(x^2)$, so you don't have to worry about any ambiguity). You will occasionally see$f^2(x)$, but I would not recommend this--$f^2(x)$more often means$f(f(x))$. The notation$f^2(x)$for squaring$f(x)$is generally used for only a few particular functions (typically ... 0 Square of a function$f(x)$can be denoted as$f^2(x)$, similar to$sin^2x$and$cos^2x\$.

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I tried putting myself in the shoes of Leibniz. And here is my perception. Which I can only attempt to explain. Consider the definition of the derivative $$f'(x) = \lim_{h\to0}\dfrac{f(x + h) - f(x)}{h}$$ Let A represent the area of a rectangle whose base is x and height is f(x). (At this time it helps to draw a concave up curve in the 1st quadrant of a ...

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