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I'm reading Boolean Algebra and Its Applications and come across this statement about Venn diagrams:

It should be remembered that such diagrams do not constitute proofs, but rather represent illustrations which make the laws seem plausible.

But I cannot distinguish a Venn diagram from just another form of notation. What is the difference, and why is a Venn digram not rigorous enough?

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    $\begingroup$ Because there may be multiple ways to draw the diagram. $\endgroup$ – Nishant Oct 19 '14 at 5:06
  • $\begingroup$ And that makes it not a proof or not notation? If the former, aren't there multiple proofs for the same theorems? And if the latter, aren't there multiple notations for the same idea? $\endgroup$ – gwg Oct 19 '14 at 5:09
  • $\begingroup$ After one has proved that any Boolean algebra is isomorphic to an algebra of sets, an argument could be made along your lines. Certainly not before. $\endgroup$ – André Nicolas Oct 19 '14 at 5:10
  • $\begingroup$ I meant that there are multiple possible non-isomorphic ways to draw the diagrams, so you'd need to draw all possible diagrams, prove that they are all possible diagrams, and show that the desired statement is true for all the diagrams. $\endgroup$ – Nishant Oct 19 '14 at 5:11
  • $\begingroup$ It seems like you can use Venn diagrams to prove statements about subsets of $\mathbb{R}^2$ $\endgroup$ – Baby Dragon Oct 19 '14 at 5:19
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It's not saying something particularly deep, just that when you actually do axiomatize these things then a formal proof will require a string of deductions from the axioms. If you want you can build things up in such a way that a (suitably formalized notion of a) Venn diagram constitutes a proof.

EDIT: Reading a bit more of the context (I can only see a limited preview from here), I believe they're trying to motivate the definition of a Boolean algebra by showing that algebras of sets satisfy the Boolean algebra axioms. So you could regard things done with Venn diagrams as proofs in an algebra of sets, but then as Andre Nicolas notes in the comments you can't use Venn diagrams to prove general facts about Boolean algebras since you don't know that every Boolean algebra can be described as an algebra of sets.

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  • $\begingroup$ I don't think a sole Venn diagram ever constitutes a proof, even if things are suitably formalized. See my answer. $\endgroup$ – goblin Oct 19 '14 at 17:13
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Actually, proofs of identities of sets by Venn diagrams are formal, although some people seem not to be aware of this. You can take a look at The truth about Venn diagrams by Ian Stewart, published in The Mathematical Gazette Vol. 60, No. 411 (Mar., 1976), pp. 47-54 (http://www.jstor.org/stable/3615644).

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While no particular Venn diagram can constitute a proof, I think that a pair of trees of Venn diagrams surely can. For example, to prove the identity $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ you could construct two trees, one for the LHS, the other for the RHS. The nodes of these trees are themselves Venn diagrams. Consider, for example, the LHS tree. It has three leaves, which are Venn diagrams denoting $A,B$ and $C$ respectively. The nodes $B$ and $C$ are joined to a third node $B \cup C$ above them, which is a Venn diagram denoting their union. We then join the nodes $A$ and $B \cup C$ to another node above them, which is a Venn diagram denoting their intersection. Construct a tree similarly for the RHS. Since the topmost Venn diagrams "look identical," this proves the identity of interest.

This is essentially a pictorial way of doing Truth Tables. The main downside is that it is limited to proving identities involving three variables.

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Venn diagram cannot (and does not) prove or disprove anything. You can tell that it gives an insight how the things are actually happening, or how a proof can be constructed.

Allow me share my thoughts a bit (the ways I like it to think):

  1. You can draw Venn diagram in uncountably different ways.Even if you prove (or disprove) a certain thing, you prove it only for some specific diagram - not for all possible Venn diagrams.
  2. What do you in Venn diagrams? You draw separate figures for the same problem. What do you call a proof, is based on comparison of two figures. These two figures represent some sets. Even if the figures look identical, it does not in any way prove that the sets are actually identical. At maximum, you can claim that apparently these two sets are identical.
  3. What proves that, a set (may be infinite) can be represented by a circle on a piece of paper?
  4. Even if you agree all these facts, you must admit that the thing is proved only for the set containing the points inside the circles, and not for all sets.

I am not from mathematics (so, my thoughts may not always be correct), but I think it may be useful to you.

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