# What does the slash mean in this set notation?

In Section 2.2, Definition 1 of this paper, it has the following:

Here, $$x,y \subseteq X$$ and $$X$$ is a set of $$d$$-dimensional vectors. $$dist(x,y)$$ stands for the Eucledian distance between $$x$$ and $$y$$.

What is the highlighted notation in points $$1)$$ and $$3)$$ ? ex: $$N \subseteq X/\{x\}$$

Can you explain those points in plain English?

• I guess it's set difference Commented Mar 2, 2019 at 19:05
• @Berci that makes sense. Please post that as an answer. BTW, I haven't seen this used anywhere to denote set difference - maybe it's just me. Commented Mar 2, 2019 at 19:11

It (probably, I can't access the full text of the paper) means "without". So $$X \setminus \{x \}$$ means "the set $$X$$ without the point $$x$$", but usually the backslash is used for that. The LaTeX command for the slash is "\setminus". It could also be quotient, for example used for groups, then $$N / G$$ means "$$N$$ modulo $$G$$".
• Yes, it means "without." Interpreting it this way is consistent with k-nearest neighbors as I know it. Read this way, the three clauses simply mean, "(1) The $k$ nearest neighbors are drawn from the original set but (of course) can't include the set itself. (2) There are $k$ nearest neighbors and (3) All the other points (beyond the given point and its k nearest neighbors) are farther away than these k nearest neighbors (or equally distant)." I can't imagine that it could possibly mean anything else, regardless of what the rest of the paper says. Commented May 24, 2022 at 15:51
Without more context I would assume that the writer used the forward slash '/' instead of the more conventional backward slash '\' to denote set difference. Thus $$A / B$$ would denote the set of elements of $$A$$ which are not in $$B$$.
The first line then translates as "$$N$$ is the set made up of the elements of $$X$$ distinct from $$x$$" while the third translates as "Given any element $$y$$ of $$N$$ and any vector $$z$$ of $$X$$ which cannot be written as the sum of an element of $$N$$ and $$x$$, the distance between $$x$$ and $$y$$ is less than or equal that one between $$x$$ and $$z$$".
More formally, $$S \setminus \{a\} := \{ x \in S \mid x \ne a \}$$.