Is the index notation $1 \leq i \neq j \leq k$ unambiguous?

Question

From the context where I found the statement $$1 \leq i \neq j \leq k,$$ the author seems to mean that

$1 \leq i \leq k$ and $1 \leq j \leq k$ and $i \neq j.$

However, strictly speaking, the given statement doesn't imply that $i \leq k.$ For example, strictly speaking, $i=5,j=3,k=4$ does satisfy the given statement. Or is there an implicit set of rules or conventions that parses the given statement as the above?

If the given statement is decidedly ambiguous, then is there a succinct and unambiguous way to express the bullet point above?

I don't think it's ambiguous, but I also think it's more standard to write, "$1\leq i,j\leq k$ with $i\neq j.$" — Adam Rubinson, Oct 18 '24 at 10:21
@AdamRubinson added a bit more of context. Particularly, if these are all binary relations wouldn't $i = 5, j = 3, k = 4$ satisfy all explicit binary statements? — Kuku, Oct 18 '24 at 11:07
@Kuku actually I think you are right. So my "standard" notation is ambiguous. — Adam Rubinson, Oct 18 '24 at 11:08
@Kan't I think where I'm coming from is that as a non-mathematician, it's these implicit norms of when is notation to be read strictly and when not to what makes one stumble from time to time. In some way, the gist of the question is to check that these are actually just 'norms' and not a mistake on my 'rule-following'. — Kuku, Oct 18 '24 at 11:13
The first thing that went through my mind after reading the title was: $i$ going from $1$ to $k$, with $j$ excluded. I imagine that it would be understood from the context that the intention was to cycle through both $i$ and $j$, but that's not the first thing I thought. There is no implicit set of rules. I prefer @AdamRubinson suggestion at this point. I would write $i\neq j$ below $1\leq i,j \leq k$ if written below a $\Sigma$ as index bounds.
WRT. your question, the meaning is absolutely indended to convey $i \leq k$ and there is no doubt about that. — Kolja, Oct 18 '24 at 11:18
@Kolja that's an additional interpretation I had not considered! — Kuku, Oct 18 '24 at 11:20
After the editing, now it's definitely clearer what you (possibly) meant, and I agree with the editor's interpretation. — Kan't, Oct 18 '24 at 11:22
Yes, this is understood from context, even if they are not 100% formal. While you argue that formally $i\leq k$ is not implied, this becomes more a game of definition and formality. One would generally not combine $\neq$ with $\leq / \geq$ because this breaks transitivity. It is only used to simplify the notation, so that one doesn't have to write the full statement $1\leq i\leq k, 1 \leq j \leq k, i \neq j$, which is 100% the indended meaning. — Kolja, Oct 18 '24 at 11:29
WRT the second interpretation - this might as well the index set of all $k$'s which satisfy some bounds for a pair $(i,j)$ only if they satisfy a constraint $i\neq j$. At this point we don't know what it is, so it depends on the context. — Kolja, Oct 18 '24 at 11:30
@Kolja If you wrap those comments into an answer I will accept it — Kuku, Oct 18 '24 at 11:33

score 3 · Accepted Answer · answered Oct 18 '24 at 11:51

The first thing that went through my mind after reading the title was: $i$ going from $1$ to $k$, with $j$ excluded. I imagine that it would be understood from the context that the intention was to cycle through both $i$ and $j$. Without context this might as well be the index set of all $k$'s that satisfy some lower bounds defined by a pair $(i,j)$ and some relation $(i\neq j)$ between them. At this point we don't know, so it is context-dependent.

In general there is no implicit set of rules.

With respect to your comment, this is understood from context, even if they are not 100% formal. While you argue that formally $i\leq k$ is not implied, this becomes more a game of definition and formality. One would generally (i.e. when being super-formal) not combine $\neq$ with $\leq / \geq$ because this breaks transitivity. This is only used to simplify the notation, so that one doesn't have to write the full statement $1≤i≤k,1≤j≤k,i≠j$, which is 100% the intended meaning.

I would write something like this in the context of a sum: $$ \sum_{\substack{1 \leq i,j \leq k \\ i \neq j}} $$

However be aware that the meaning is absolutely intended to convey $i \leq k$ and there is no doubt about that.

Thanks for the writeup. Continuing on the hairsplitting, I assume there is also no formal convention regarding the usage of the comma in indices, no? Would $1 \leq i,j \leq k$ ever be read as $1 \leq i$ and $j \leq k$, or is it always to convey some sort of tuple or set to which binary relations apply element-wise, i.e. $1 \leq {i,j} \leq k$? — Kuku, Oct 18 '24 at 12:29
$a_1,a_2,\dotsc,a_n \geq c$ is understood as $a_i \geq c$ for all $i$ — Martin Brandenburg, Oct 18 '24 at 12:32

Is the index notation $1 \leq i \neq j \leq k$ unambiguous?

1 Answers1