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

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    $\begingroup$ That looks like a horrible abuse of $\cap$; surely, it must be defined earlier in the work? I would note that, for sets, the claimed relation is not an equivalence relation. For example, let $x_1=\{a,b,v\}$, $x_2=\{b,c,v\}$ and $y=\{v\}$. Then $x_1\mathcal{R}_v y$ and $y \mathcal{R}_v x_2$, but $x_1\cap x_2 = \{b,v\}\notin \{x_1, x_2,v\}$ so $x_1 \mathcal{R}_v x_2$ does not hold and the relation is not transitive. $\endgroup$ – David Richerby Mar 26 at 12:10
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    $\begingroup$ Please edit your question to include a full citation to the material you're quoting. $\endgroup$ – David Richerby Mar 26 at 12:16
  • $\begingroup$ DavidRicherby and dkaeae, thank you for the posting tips. I have updated the question so that the equations are written in Latex and a source is included. $\endgroup$ – chillsauce Mar 26 at 18:36
  • $\begingroup$ @DavidRicherby Unfortunately I could not find any earlier definition of ∩ in the paper. I did find that the files $x_i$ are Boolean variables if that helps. Doing a cursory Google search, I couldn't find a common definition of ∩ for Boolean variables. $\endgroup$ – chillsauce Mar 26 at 19:14

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.

  • $\begingroup$ Thank you so much for your answer. $\endgroup$ – chillsauce Mar 26 at 21:02
  • $\begingroup$ Actually, for sets, $x_1\cap x_2 \in\{x_1,x_2,v\}$ means that $x_1\subseteq x_2$, $x_2\subseteq x_1$ or $x_1\cap x_2=v$. That's a slightly longer condition than the one you've wrote so maaaaaaybe somebody would write it that way. People often think that kind of "assembly language hacking" of notation is a good idea: for example, we had a question the other day where somebody had written "The string contains more 1s than 0s" as "$2\sum_{i=1}^{|x|}x_i>|x|$". $\endgroup$ – David Richerby Mar 26 at 22:03
  • $\begingroup$ That last example is AMAZING! Haha! In line with what you said, I think the author must have have meant for the $x_i$ variables to act like sets of substrings of the bitstring of each file when the ∩ operator is applied. It still confuses me that the paper states that they are Boolean variables not too far before though. $\endgroup$ – chillsauce Mar 26 at 23:07
  • $\begingroup$ @DavidRicherby, oh, gosh yeah! And good point about why someone might have written that -- it's not quite as far-fetched as I originally thought -- so that part of my argument no longer looks persuasive to me. Thanks for the correction and good counterargument. Thank you. $\endgroup$ – D.W. Mar 27 at 0:19

This is some poorly written textbook. It should distinguish a variable and its realization (i.e. its value). I would rewrite as the following. Let $f_1$ and $f_2$ be some file variables. $x_1$ and $x_2$ be values of the files. Then $$f_1 R_v f_2 \text{ if } f_1 = f_2 \in \{x_1, x_2, v\}$$

In words (using the original notation), it probably means something like below.

The relation $x_1 R_v x_2$ holds if either of the following is true.

  1. $x_1$ and $x_2$ contains content of $x_1$
  2. $x_1$ and $x_2$ contains content of $x_2$
  3. $x_1$ and $x_2$ contains content of the virus $v$
  • $\begingroup$ I’ve rewritten to make it easier to understand. Remember, $f_i$ are variables and $x_i$ are values of the variable like integers. $\endgroup$ – James Parker Mar 26 at 22:01

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