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 $\pm\sqrt{a}$.


4

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 $f``A$. 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 use $i = 0$ 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| \...


1

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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