"For all" notation with inequalities

Question

I am working on an assignment and have a question regarding the use of the $\forall$ symbol with inequalities. Here is my current expression:

$$|D_n| = 0 \space\space \forall \space\space n \space\space s.t. \space -3>n>3 \space, \space n \in \mathbb{Z}$$

Objective: I want to say that the magnitude of $D_n$ is equal to zero if n is less than $-3$ or greater than $3$. My concern comes from the inequality itself, I have an inclination that I may have to break the statement in two, one for $n<-3$ and one for $n > 3$.

Additionally, I have some confusion regarding the notation for "such that", I have seen conflicting statements online regarding the use of $\ni$, or $\ni'$ as a notation for "such that". I undestand that using "$s.t.$" is less ambiguous, however, I am curious nonetheless.

I appraciate your time and help greatly! My apologies if this question is rather amateur, I am still learning!

Answer 1

Your usage of for all, and such that, is fine; the inequality is not fine, since it suggests $-3\gt 3$ which is not so. There are a few ways to write what you’re saying in a better way:

$$\forall n\in\{-4,-5,\cdots\}\cup\{4,5,\cdots\},\,|D_n|=0$$

Or:

$$\forall n\in\Bbb Z,\,|n|\gt3,\,|D_n|=0$$

Or:

$$\forall n\in\Bbb Z\setminus\{-3,-2,\cdots,3\},\,|D_n|=0$$

Or:

$$\forall n\in\Bbb Z:n\lt-3\vee n\gt3,\,|D_n|=0$$

Where \vee $\vee$ is the logical OR.

Notice that it is not even necessary (but of course you can) to write “such that”. Sometimes we abbreviate such that, or imply it grammatically, by using a colon.

The $\ni$ notation can be used for “such that”, but this is not very common in my experience. It is much more common to use it to denote set membership, as $\in$ does, except that the author is trying to emphasise the set which comes first, or perhaps that it makes more sense to let things flow in the $\ni$ direction than the $\in$.

Example:

You might want to say that $U$ is a set containing a point $x_0$, but here you are maybe letting $U$ be an arbitrary set with some conditions plus the condition that $x_0\in U$. However, in the sentence structure it can be more natural to write (e.g.):

“There exists an open $U\ni x_0$”

As opposed to:

“There exists an open $U$, with $x_0\in U$”

Another use case is in function declaration, where the author does not care to give the map a name as such since this is not the important; they rather want to show what this map will do and what properties it has. That is, it’s common to write $f(x)=x^2$ as a function definition, especially lower down the school, but more abstractly an author might write:

“Since the map: $$\Bbb R\ni x\mapsto x^2\in\Bbb R$$Is continuous...”

And the rest of the sentence would maybe make reference to the map, to the function, but it was never necessary to give the function a name (e.g. $f$). This notation also shows where $x$ comes from (what kind of number/object it is) and it can flow better to use $\ni$ there rather than:

“Since the map: $$\Bbb R\to\Bbb R,\,x\mapsto x^2$$Is continuous...”

As this is maybe longer or not preferable for some authors.