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74

It is perfectly fine to write $|A|>0$. However, the simplest and most common way to write this in symbols would be $$A\neq\emptyset.$$ Note that you don't want to write $|A|\neq \emptyset$, as it is $A$ itself which you are saying is not the empty set, rather than the cardinality of $A$. (The standard symbol in mathematics for "not equal" is $\neq$, ...


55

None of the answers mention that professional mathematicians don't specially go out of the way to convert everything to symbols. "$A$ is non-empty" is indeed the most common way to express the statement. Furthermore, for complicated structures it is almost always expressed this way, such as: Given any non-empty chain of fields ordered by inclusion, their ...


10

$A \neq \varnothing$ [LaTeX: A \neq \varnothing]


6

$A \neq \emptyset$ Thats how it is commonly written


6

Yes, there is a standard notation, namely $p^e\mid\mid n$, which says that $e$ is the largest power of $p$ which divides $n$. Reference: Martin Aigner, Number Theory. Edit: For more advanced purposes, like $p$-adic numbers etc., a common notation is also $\nu_p(n)$, which also then appears in more elementary context. For elementary number theory I have ...


6

Strictly speaking "will this notation I like ever catch on?" Is inappropriate for this site. But we can still weigh the pros and cons of the suggestion and mention any variants we know. Cons: Introduces yet another symbol to memorize weak improvement over the obvious alternative Rarely do you talk enough about irrationals collectively to warrant special ...


5

It's commonly used notation, so yes. And even if it wasn't, you can always define notation in the beginning, and then use it. In this case, you'd see something like "For brevity, we will use $x,y,\ldots>0$ to mean $x>0$, $y>0$, etc. in the rest of the article."


5

I always do that, however be careful if you want to write $a < x < b$ and $a < y < b$, since $a < x,y < b$ is ambiguous. One would have to write $x,y \in (a,b)$.


5

This can be seen as a special case of the more general concept of valuations (on discrete valuation rings). A common notation in that context, which is quite convenient also here is $\nu_p(n)$.


5

How about: $(\exists x)\, x\in A$? Alternatively, one could say that $A$ is inhabited. This usage avoids needless negation which is problematic constructively speaking, and is common in constructive mathematics.


4

It's not standard notation. I would not count on a reader correctly understanding it, unless you explain it. You could introduce it as your own special notation, but unless you feel it makes a huge difference in the clarity or conciseness of your work, I would not do so. Readers tend to get annoyed by gratuitous use of non-standard notation. Don't use it ...


4

You ask: "Is it written well, according to the laws of mathematical language?" There aren't really laws, just guidelines. Mathematics isn't a programming language. So I agree with J.M.'s comment 100%. But, to make the meaning of the notation a little, clearer, you might want to write something more along the lines of $$\cot\alpha\left\{\begin{matrix}+\...


4

It is common to use $:=$ when you are defining something. That way you communicate that it isn't a formula that is derived, but something that is defined. For example, I might say that the velocity of a particle is $$ v := \frac{d}{dt} s(t) $$ where $s(t)$ is the position. By using $:=$ I have told the reader that $v$ is defined as the derivative of the ...


3

Pick any $M$ within the real numbers. Make it as big as you want. Then a sequence $s_0, s_1, s_2, ...$ diverges to positive infinity if there exists some number $N$ such that $s_{N+1}, s_{N+2}, s_{N+3}, ...$ (in other words, all $s_n$ beyond $s_N$) are all greater than $M$. This value of $N$ will depend on the particular value of $M$, so it will be in ...


3

Capital N and M in these sort of definitions tend to represent large numbers (including extremely negative). $\epsilon$ and $\delta$ represent numbers near zero. "For every $M$ in $\mathbb R$ there exists a number $N$ such that $n>N$ implies that $s_n>M$." For every $M$... that means every. Since it is captialized, think big. No bigger than that. ...


3

While there is a lot of context missing from the question, perhaps it is $T^* \circ T$, where $T^*$ is the adjoint of $T$? That is easily seen to be self-adjoint and with positive eigenvalues: Since $(AB)^*=B^* A^*$ and $A^{**}=A$, we have that $(T^*T)=T^*T^{**}=T^*T$. Further, since if $T*Tx=\lambda x$ and $x\neq 0$, then $$\lambda = \langle T^*Tx,x \...


3

When the biconditional is legitimate (i.e., when there really is equivalence between the statements), it never hurts to indicate so, and IMO it's always a good idea to use it. This way you can tell if your chain of reasoning is totally reversible, and if it's not reversible, you know which step is the culprit. Even if the result you're trying to establish ...


3

Let $A$ denote a set. In constructive mathematics, there's a difference between the statements '$A$ is non-empty,' which is defined to mean that $A$ is not isomorphic to $\emptyset$, and '$A$ is inhabited,' which is defined to mean that $A$ has at least one element, i.e. $\exists a \in A(\mathrm{True})$. Thus, depending on their standpoint and interests, a ...


3

$(0,1)$ does not stand for an ordered pair, but for “the interval $(0,1)$”: $$ (0,1)=\{x\in\mathbb{R}\mid 0<x<1\} $$ so you are defining the set of all real functions $f$ defined over the interval $(0,1)$ which are differentiable, with continuous derivative. The notation $f\colon (0,1)\to\mathbb{R}$ denotes a function with real values defined over the ...


2

The first interpretation is correct. For the second one, you would write $$\{x\in \mathbb R ,\ 5<x<7\},$$ or simply $$\{5<x<7\},$$ or (thanks to a comment) : $$x\in(5,7).$$


2

min! is the argmin, i.e., $f(x)=\min!$ is identical to $\arg \min f(x)$. See also http://mathoverflow.net/questions/182112/a-question-about-some-notation-involving-the-exclamation-mark and the link there, which discusses your example.


2

Yes, you can do that. Either way is fine, but, of course, the former is more concise.


2

The "=" is equality. The case you give is definition. It is also sometimes used as redefinition, e.g. x := x + 1 for a programming language.


1

The notation $A=B$ means $A$ is equal to $B$. The notation $A:=B$ means "Let $A=B$." It means you're saying what you will mean when you write $A$. I suspect the $\text{“}{:=}\text{''}$ notation hasn't existed for more than about a half a century, so it's brand-new.


1

This notation is borrowed from Computer science and means (in Computer science) ‘takes the value…’. I prefer the more explicit old style $$f'(a) \stackrel{\text{def}}{=}\mkern1mu \lim_{h\to 0} .\frac{f(a+h)-f(a)}{h}$$


1

$E = f(n) + \dfrac{|n - 1/2|}{1/2 - n}$


1

Are you familiar with universal quantifiers and existential quantifiers? This might help you a little because it gives us a framework for thinking about these kinds of statements. For example, this statement is: $\forall M \in \Bbb{R}$ --> This is the same as saying "choose any $M \in \Bbb{R}$" $\exists N \in \Bbb{N}$ --> This is the same as saying "there ...


1

It is just an index. It means, that for any $s \geq r+1$, such a factorization exists and the factors are called $g_i^{(s)}$, since they of course depend on $s$.


1

You can say something like: Let $\displaystyle x=\sum_{i\mathop=0}^n a_i \times 10^i$. Then the sum of digits is $\displaystyle \sum_{i\mathop=0}^n a_i$ You can also define a "sum of digits" function recursively (for positive integer only): $\text{sum_of_digits}(x)=\begin{cases}x \text{ , if } x<10\\ \text{sum_of_digits}(\left\lfloor\dfrac x{10}\right\...



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