What does the slash mean in this set notation?

Question

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?

Answer 1

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 $G$".

Answer 2

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 $z$".

Answer 3

More formally, $S \setminus \{a\} := \{ x \in S \mid x \ne a \}$.