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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?

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marked as duplicate by Lord_Farin, Rick Decker, Amzoti, Mark Bennet, Asaf Karagila Jun 7 '13 at 14:28

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2… – Belgi Jun 7 '13 at 12:25
@Belgi: thanks for the heads-up... I probably should delete this post, because I think it is very much a duplicate of the one you linked to. – kjo Jun 7 '13 at 12:35

2 Answers 2

up vote 3 down vote accepted

In a Venn diagram, if membership in a set $A$ implies membership in a set $B$, then indeed $A \subseteq B$.

There is a traditional notation in which "$A \supset B$" is used to mean "$A \to B$". That notation re-uses the superset symbol but it is not directly about supersets. The $\supset$ there is derived from notation of Peano in which he used a backwards "C". This traditional notation does not use $\supseteq$ however; the horizontal lines are a more modern invention.

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What you would like to do is, in some sense, map $A$ by any contravariant operator (shortly, that is the one which changes direction of ordering). For example the left side of $\color{blue}{\text{implication}}$ is contravariant (different color to avoid confusion), and gives the following interpretation:

$$A \rightarrow B \text{ implies } [A] \supseteq [B],$$

where $[A]$ denotes the collection of all terms $\color{blue}{\text{implied}}$ by $A$.

I hope this helps ;-)

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