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## Hot answers tagged notation

4

By definition (when $a> 0$), $a^{1/2}$ and $\sqrt a$ both mean the positive square root of $x$. If you wanted to include both numbers which square to give $a$, you would need to write $\pm a^{1/2}$ or $\pm\sqrt{a}$.

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The question has already been answered, but here is a little context: the notation $Y^X$ is a little odd to denote the set of functions $X \rightarrow Y$. The origin of this notation lies in combinatorics, and more generally set theory. Let $[n]$ be the set $\{1,\dots, n\}$. It is a classic motivational exercise in combinatorics to count the size of the set $... 4 The answer depends on context. If you found this notation in reference to a partial differential equation or similar real analysis setting, it most likely means that$f \in C^2$and$f''$is Lipschitz. This is a special case of the notation$f \in C^{2,\alpha}$to mean$f \in C^2$and$f''$is$\alpha$-Holder continuous. Note that$\alpha=1$in the ... 3 It means that if$f\in \mathbb{F} ^S$then$f$is a function from the set$S$to a field (think$\mathbb{R}$or$\mathbb{C}$for example),$f:S\to\mathbb{F} $. Also,$\mathbb{F} ^S$, with the sum (sum of functions) and product by scalar (scalar times function) as defined in 1.23 is a vector space. 2 In lattice theory, this is nothing but the Heyting arrow in the lattice$\mathcal P(S)$of subsets of a set$S$: $$(A\succ B) = A^c\cup B.$$ More generally, if$L$is a meet-semilattice (meaning a partially ordered set with finite infima$\wedge$), and$a,b\in L$are any two elements of$L$. An implication from$a$to$b$is an element$(a\succ b)\in L$such ... 2$4^{1/2}$is evaluated to be$2$. Fundamentally, the function$f(x)=x^{1/2}$has a range of$[0,\infty)$. In comparison, if we were looking for the solutions to the equation $$x^2=4$$ instead, then in solving the quadratic equation we would get$x=\pm2$. 2 I think this is not uncommon, although you more often see the subscript for defining a sequence, e.g.$(a_n)_{n\geq 1}$. Alternatives, which are more usual for sets, would be $$\{(\mathbf{x}_i,y_i)\mid 1\leq i\leq N\}\quad\text{or}\quad\{(\mathbf{x}_i,y_i): 1\leq i\leq N\}.$$ (I might write$i\in[N]$, with$[N]$meaning$\{1,\ldots,N\}$, but I'm not sure how ... 2 Some common notations for the converse of a relation$R$:$R^\text{T}$,$\breve{R}$,$\check{R}$. Now these aren't quite what you're asking for. As you note, if$f:X\rightarrow Y$, then the function mapping$P(X)\rightarrow P(Y)$(powersets) is different. However, another standard notation for$f(A)$, with$A\subseteq X$, is$fA. So you might write $$\... 2 In general, you want to make sure that you start/stop the sum with the same terms. For example, this sum starts from 7 and ends at 20:$$\sum_{i=1}^{14}(i+6).$$If we want i to start from 3 (say), then we need to adjust the function inside the sum, and the upper limit, so that it still starts from 7 and ends at 20:$$\sum_{i=3}^{16}(i+4).$$In general, ... 2 Approximately equal to. For instance, \pi≐3. Usually, people use a squiggly equality signs for approximations, but ≐ means the same thing. ≐ means approximately equal to. 2 I would say it’s a member of (\Bbb R^2)^{2\times 2}. If A is some ring, A^{2\times 2} is the set of two by two matrices with elements in A. 2 The symbol has no strong standard meaning. Wikipedia says Symbols used to denote items that are approximately equal include the following: \doteq (U+2250), which can also be used to represent the approach of a variable to a limit Simple Wikipedia says There are several symbols that can be used to say items are "approximately the same," "... 1 [https://web.stanford.edu/~boyd/cvxbook/][2] Convex Optimization, Stephen Boyd and Lieven Vandenberghe 1 Update: This answer lists sources which use various symbols for this set operation. (No strong consensus, but at least there is some precedent in the literature.) I have not been able to find any literature containing a symbol for A \cup B^\complement, so I'm tentatively saying the answer here is: no, there is no precedent for such a symbol. Here are some ... 1 There is not perfect choice of notation. Look into the lecture notes or ask your tutor/other students. I like this notation for group actions: For a left action of g \in G acting on an element x you could write$$ g \rhd x. $$To differentiate between different group actions, you could introduce for example \rhd_A and \rhd_B to make clear which group ... 1 An expression or function evaluation has only one value (or "answer") - or perhaps none in case it is not defined. Even though the equation x^2=9 has two real solutions x=3 and x=-3, the expression \sqrt 9 has only one value 3. What "is" a^b, anyway? In order to know what is meant by a^b, we need to define the expression a^... 1 In the most general setting, considering an arbitrary set A, a magma (M, \cdot) -- which is a structure obtained by equipping the arbitrary set M with an arbitrary binary operation \cdot \colon M \times M \to M -- and two maps f, g \colon A \to M one can define the product of the two maps as the map given by:$$\begin{align} fg \colon A &\to M \... 1 Just usei = 0$in your image. Then for 2), 3), 4) you just add$0$to your sum. In 1) you add another c, so it changes the summation but it is easy to think about it. Try it yourself. For$i \geq 2$you can write it differently, e.g. $$\sum\limits_{i=2}^n c = \sum\limits_{i=1}^n c - \sum\limits_{i=1}^1 c = cn - c = (n-1)c$$ 1 First of all: there are no problems in "not being good at math". If you want a math solution, them you're free to come at SE and ask. For your question, we can start by$x_0 = h$. Then, we define$f(x)$(or$f\circ x$) as whatever is in the loop:$f\circ x = k\cdot x$. As the result is the end of the iteration of the loop (with start at$x_0$), we ... 1 Initially,$x=h$. After one iteration,$x'=kx=kh$. After two iterations,$x''=kx'=kkh=k^2h$. More generally, after$n$iterations, $$x^{(n)}=k^nh.$$ In a more mathematical language, you have the recurrent sequence defined by $$x_0=h,\\x_n=kx_{n-1}.$$ 1 It means almost everywhere with respect to the measure$\mu$: for short,$\mu$-almost-everywhere. Id est, that the set of points where the condition fails is contained in a measurable set$S$such that$\mu(S)=0$. 1 Your sentence is weaker than what the limit is saying. As an illustration,$-1/x$on$x \in (0,\infty)$has all its values negative, but its limit infimum is zero. It is generally useful to speak in terms of "tail"s when talking about limits infimum and supremum. Here, a "tail" is: for each choice of$u_0$, restrict the limit as$|u| \...

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You start out wanting $\frac{\partial}{\partial x}f(xy,x^2-2)$. There is this function $f$ of two variables that I might call $u$ and $v$: $f=f(u,v)$. Here, $u$ is composed with $xy$ and $v$ with $x^2-2$. The chain rule says: \begin{align} \frac{\partial}{\partial x}f(xy,x^2-2) &=\frac{\partial f}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\...

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