What does the intersection symbol (∩) mean when applied to two non-set elements? I came across a piece of literature in which I saw the intersection symbol (∩) being used on two non-set elements in the definition of an equivalence relation; I have posted it below for reference. The elements in question are meant to represent different files on a computer. What could that intersection symbol mean in this context?


  Let $x_1$ and $x_2$ be two files and let v be a given computer virus. We then define the equivalence relation $R_v$ as follows:
  
  $x_{1}\!\mathit{R_{v}}\,x_{2} \;\; if \;\; x_{1} \cap x_{2} \in \left \{ x_{1},x_{2},v \right \}.$
  
  This is an equivalence relation and any equivalence class of a
  given element $x$ is defined by $C(x) = \left \{ y |y\in S \;\; and \;\; x \mathit{R_v} y \right \}$.
  The class $C(v)$ contains every file infected by $v$. Every class
  which is a singleton contains an uninfected file.


This text appears in the paper just before the excerpt above and may provide more context:


  Most of self-reproducing codes that exist at the present time
  are worms and thus a single copy of the malware is present
  in the system. But it is not the case as far as virus are considered
  (many copies exist in the system at the same time). In
  order for our model to be general, we will consider that all the
  different copies of a malicious codes are in fact a single one,
  e.g. the viral code. In the very special case of k-ary viruses,
  the viral code is the disjoint union of k different files.
  
  From a mathematical point of view, it is equivalent to consider
  the following equivalence relation, which is defined on
  a set S (the file system).


And even earlier, the text the author describes the $x_i$ variables as boolean variables:


  In order to define
  the working context, let is consider an operating system containing
  $n$ files ($n$ is of arbitrary size). These are all possible
  files that exist or may exist in the system at a given time.
  Each of these files are described by a Boolean variable $x_i$ ,
  $i = 1, 2, . . . , n$. No particular assumptions is made about the
  status of any of these files (executable or not, data. . .).


Source for the paper: https://link.springer.com/article/10.1007/s11416-007-0044-2
Eh.  $\cap$ doesn't mean anything on non-set variables.  Looking at the paper, that part of the paper doesn't make any sense to me either.  The paper says $x_1,x_2$ are Boolean variables, but the notation $x_1 \cap x_2$ is not well-defined on two Boolean variables; you can't take the set-intersection of two Boolean variables.  If $x_1,x_2$ were sets somehow (which the paper says they aren't), then the meaning of $x_1 \cap x_2 \in \{x_1,x_2,v\}$ is that either $x_1 \subseteq x_2$ or $x_2 \subseteq x_1$ or $x_1 \cap x_2 = v$.

So it's anyone's guess what that formula was supposed to represent.  If you absolutely must understand the paper, you'll probably have to read the surrounding context to guess what the intention was.  Better yet, find a better paper to read, if you can.  In any case, the reason you are confused is because what's written is confusing and seems like mathematical gibberish to me.  It happens; no one is perfect.  Don't assume that just because a paper is published in a journal, that what's written there will necessarily be correct or useful or meaningful.