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Textbook and online resources say something along the following lines:

$R$ is the subset relation where $(A, B) \in R$ if and only if $A, B \subseteq C$ and $A \subseteq B$.

Is this also equivalent to

$R$ is the subset relation where $(A, B) \in R$ if and only if $A \subseteq B \subseteq C$.

I think they are, but I haven't come across any that states the definition in the latter way.

I have tried a making up a few sets $A, B, C$ such that $A, B \subseteq C$ and $A \subseteq B$. So far, all of them also satisfy $A \subseteq B \subseteq C$.

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  • $\begingroup$ You're assuming that $B \subset C$ and $A \subset B$ in either case. They are exactly the same. $\endgroup$
    – user296602
    Apr 17, 2018 at 17:37
  • $\begingroup$ I'm not entirely sure how $C$ fits into the definition (is $C$ a universal superset?) but the statement 1) $A\subset C$, $B \subset C$ and $A\subset B$. and statement 2) $A \subset B \subset C$ are exactly the same thing. If statement 1) then $A \subset B$ and $B \subset C$ so that means $A \subset B \subset C$... or statement 2. Is statement 2) If $A \subset B$ and $B \subset $ then it follows that $A \subset C$ ... so statement 1. $\endgroup$
    – fleablood
    Apr 17, 2018 at 21:49
  • $\begingroup$ I think what is going on is that to define any relationship between subsets of $C$ it is understood that $A, B \subset C$ because that is the "universe" we are talking about. We are only talking about relationships between two subsets $A,B$ of $C$. The relationship itself is that $A \subset B$. Since $B \subset C$ this is simply the same thing as stating $A \subset B \subset C$. Conversely, if you have a statement $A \subset B\subset C$ the we know $A \subset C$ that can be inferred, but it goes without saying. $\endgroup$
    – fleablood
    Apr 17, 2018 at 21:54
  • $\begingroup$ @fleablood I copied the first snippet directly from the textbook. $C$ does seem to be a universal superset $\endgroup$
    – wybkqqnob
    Apr 17, 2018 at 21:54
  • $\begingroup$ I wouldn't lose sleep over this. But for thouroughness sake, could you give a reference to the textbook, and a reference to the site that defined things otherwise? $\endgroup$
    – fleablood
    Apr 17, 2018 at 21:56

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