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I know that many people like to think of elementary logic in terms of Venn diagrams, i.e., elementary set theory. I have never found this representation useful, because I can never remember whether implication is supposed to be represented by the relation "contains" or to the relation "is contained in". IOW, do we represent $A \rightarrow B$ as $A \supseteq B$ or as $A\subseteq B\;$?
For all I know, both representations could result in useful interpretations, depending on the situation.
Whenever I come across an exposition that resorts to such a representation of logic through Venn diagrams, for some reason that is (to me at least) very obscure, my initial gut-reaction is that $A \rightarrow B$ corresponds to $A \supseteq B$. This is annoying, because I eventually come to realize that my instinct is wrong: the intended representation is the one where $A \rightarrow B$ corresponds to $A \subseteq B$.
(I want to stress that I have no problem at all understanding the correspondence between $A \rightarrow B$ and $A \subseteq B$. My problem is only that this correspondence is not at all intuitive: I always need to think it through, or "compute" it, so to speak, and this makes this representation more of a hindrance than an aid to my thinking.)
I rack my brains trying to figure out why my instinct here is so backwards (and apparently incurably so).
The only possible explanation I can come up with (and I'm definitely "grasping at straws" here) is that maybe there is some situation in which the representation "$A \rightarrow B$ is $A\supseteq B\,$" is actually useful and used, and maybe I learned it first somehow?
My question is: does anyone know of a reasonably common application of representing the implication $A \rightarrow B$ as $A \supseteq B\;$? Conversely, does anyone know of a good reason for why this representation would rarely, if ever, be useful?