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6

Most authors use $\rightarrow$ for a general function to denote that it is mapping elements from its specified domain to its specified codomain. Other, very similar notation is sometimes used for functions with special properties. For example, $\twoheadrightarrow$ is sometimes used to emphasize that the function is surjective, and $\rightarrowtail$ to ...


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In this context, it seems to be the set of all mappings, which you can see because on the right of the equation you have $A \rightarrow (B \rightarrow C)$, and this wouldn't really make sense if $B \rightarrow C$ were a single mapping. I think $B \rightarrow C$ is usually written $C^B$. This 'equation' is called currying a function, where if we take a ...


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On page 15, around the middle you can find the following: $2.22$. Definition. For any two sets $A$, $B$, $$(A\to B)=_{df}\{f\mid f\colon A\to B\}=\text{ the set of all functions from }A\text{ to }B.$$ So $(A\times B\to C)=_c(A\to(B\to C))$ simply means that the set of functions from $A\times B$ to $C$ has the same cardinality as the set of functions ...


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If you are asking about the equality $$E_p\left[\log_2 \frac1{p_x}\right] = -\sum_x p_x\log_2 p_x$$ then it is just the definition of expected value and using $\log_2\frac1x=-\log_2x$. (Which, of course, works for logarithms at any base, not just base two.)


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$S[x]$ is the ideal of polynomials with even coefficients. Generally if $I \subset R$ is an ideal and $M$ is a module, the $IM$ is the submodule of $M$ generated by elements of the form $im$, where $i \in I$ and $m \in M$. Sometimes you can use shortcuts, for example $I R[x] =: I[x]$. It's not maximal in $\mathbb{Z}[x]$: the quotient $\mathbb{Z}[x] / S[x]$ ...


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Concerning https://de.wikipedia.org/wiki/Elementzeichen#Geschichte it was first used 1889 in the work Arithmetices principia nova methodo exposita (page X): „Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b“ This means something like (my translation which isn't the best!): The symbol ϵ means is. So a ϵ b has to be read as a is a b Here ...


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The gist of the OP's explanation of why the "cancellation" of $\partial f$'s should not be allowed (and does not work) is correct, but something more can be said. The partial derivative $\partial f/\partial x$ is the rate at which $f$ changes with respect to change in $x$, but while holding y constant. Similarly the definition of $\partial f/\partial y$ ...


2

Given two matrices $M_{m\times n},N_{m\times p}$, there are two ways to interpret the entity $\begin{bmatrix} M & N \end{bmatrix}$. One is the $m\times (n+p)$ matrix whose $(i,j)$ entry is $\begin{cases} (M)_{(i,j)}, &\text{if }j\leq n\\ (N)_{(i, j-n)}, &\text{if }j\ge n+1\end{cases}$. In this case I'd rather denote the matrix described above ...


2

In general $A^{S}$ denotes the set of functions with domain $S$ and codomain $A$. In the special case $2^{S}$ where $2$ stands for $\left\{ 0,1\right\} $ you are dealing with characteristic functions so that there is a one-to-one correspondence between the elements of $2^{S}$ and the subsets of $S$.


2

The notation $A \subset \subset B$ implicitly presupposes that $A$ and $B$ are subspaces of some Hausdorff space $X$, for example $X=\mathbb C^n$. In that case it means that the closure $\overline A$ of $A$ in $X$ is a compact subset of $B$. This is the usual interpretation, for example the one adopted by Hörmander's classic An introduction to complex ...


1

I have only seen $\frac{d}{dx}h(x)=\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$. Even with the prime notation, you don't say $h'=f'+g'$, unless it is for shorthand for scrap work. If it is being graded, I would always specify that the function is some function of a variable- in this case $x$.


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Honestly, you should also write $$h'=f'+g',$$ since you take derivative of functions, and not of values of functions. Anyway, Leibniz' notation emphasizes the variable (you can't write $\frac{d}{dx} \sin t = \cos t$), so you might prefer to write $\frac{d}{dx}h(x) = \frac{d}{dx} f(x) + \frac{d}{dx}g(x)$. My favorite notation remains $Dh=Df+Dg$, which is ...


1

It means, precisely, the set of real analytic functions. The idea of the notation is that $\omega$ is the ordinal following all the finite ordinals, it is larger than all of them: being analitic is a bit more than being differentiable to all orders. I doubt one can trace who first used the notation, really. (Is it at all interesting?)


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In a directed graph, the Cartesian product notation is spot on. However, in an undirected graph, I don't like it. It indicates $(v_{1}, v_{2}) \neq (v_{2}, v_{1})$. Of course, notationally, we can wave our hands based on context. I prefer $E = \{ (v_{1}, v_{2}) : v_{1}, v_{2} \in V, v_{1} \neq v_{2} \}$. This also takes care of the diagonal entries, as we ...


1

As others have mentioned, the cartesian product would include both $(u,v)$ and $(v,u)$, which is not desirable for an undirected graph. The cartesian product also includes $(v,v)$, which is not desirable for simple graphs. For a simple undirected graph with vertex set $\cal V$ and edge set $\cal E$, you could instead define $\cal E$ as a subset of ...


1

If you look at row of people lined up in front of a mirror, not only will their mirror images appear in the opposite order (the image of the person closest to the mirror comes first) but also the image of each individual person will be a mirror image (if facing towards the mirror, the mirror image will be facing out of the mirror) just as if the person were ...


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what bothers me is the first part it is not specified what function we are taking the derivative Assume you have two differentiable functions $f:I\rightarrow \mathbb{R} $ and $g:J\rightarrow \mathbb{R} $ such that $f(I)\subset J$, where $I$ and $J$ are two intervals $\varnothing \neq I\subset \mathbb{R} $ and $\varnothing \neq J\subset \mathbb{R} ...


1

Some further generality may be informative: sometimes $A^B$ is used to indicate the set of all functions $B\to A$. When one of these two sets is replaced by a natural number, it indicates an arbitrary indexing set with that cardinality. So one can think of $\mathcal{S}^n$ as the set of functions $\{1,\dotsc,n\}\to\mathcal{S}$. But each such function $f$ can ...


1

Before looking it up just now, I didn't realize you couldn't have duplicate elements in a set. Intuitively "$A\cup B\cup C = A \cup B - A\cap B - A\cap C$" made a lot of sense to me because of this. If you were under the misconception that sets could have multiple copies of the same element (as I was just a couple of minutes ago), then it's not much of a ...


1

Most differences are just derived directly from the language. Sine in Portuguese is "seno" so "sen" makes logical sense. In Russian, tangent is тангенс (pronounced as "tangens"). Just use google translate you you should be able to figure out. Some countries use the comma instead of dot to denote decimals: i.e 0,5 instead of 0.5


1

This is the smallest subgroup that contains both $(12)(34)$ and $(234)$. This object is well defined as the intersection of all subgroups that contains $(12)(34)$ and $(234)$ because intersection of subgroups is a subgroup. Another description: $$ H= \{ a_1a_2\cdots a_n: n\in\Bbb N, \forall k\ \ a_k\in\{ (12)(34),(234) \} \} $$


1

As you suggest in your own question, there is in fact no contradiction in Leibniz's notation, contrary to persistent popular belief. Of course, one needs to distinguish carefully between partial derivatives and derivatives in the notation, as you did. On an even more basic level, the famous "inconsistency" of working your way from $y=x^2$ to $dy=2xdx$ is ...


1

This notation doesn’t only describe a system of linear inequalities but a linear optimization problem. So line 1 is the objective function and the other lines are a system of linear inequalities that describe the feasible region. By the way, this description is a use of set builder notation (with cleverly sized braces): The problem is to find the minimum of ...


1

Usually, the symbols $A \times B \to C$ denote a specific mapping from the Cartesian product of $A$ and $B$ to $C$. It could be that the author means that setting parentheses around it should denote the set of all mappings $A \times B \to C$. That is, in more standard notation $(A \times B \to C) = \hom(A\times B, C)$. It is a standard fact that $\hom(A ...


1

The notation $X \to Y$ usually denotes the type of functions which has domain $X$ and codomain $Y$. In context of set theory, that would be just the set of all functions from $X$ to $Y$, sometimes written also as $Y^X$ or ${}^XY$ to avoid order confusion (note the inversion in the first). Hence, using the alternate notation, $A \times B \to C$ is $C^{A ...


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The question is ill-posed, as there was no concept of function before Bernoulli and Euler in anything approaching the modern sense. Since there was no concept there could not have been a notation for it. In the 17th century mathematicians mainly worked with curves defined by an equation, and studied there properties. This does not require an abstract notion ...


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The diagrams are a way of describing a group generated by reflections. Any collection of reflections (in Euclidean space, say) will generate a group. To know what this group is like, you need to know more than just how many generators there are: you need to know the relationships between the generators. The Coxeter diagram tells you that information. There ...


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It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $\Theta$" (see p. 448), as in $$ f \asymp g \iff \exists\, C,D>0 : C|g| \leq |f| \leq D|g|, $$ but I read a paper recently where it was instead defined to mean the same thing as $\sim$ ...



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