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Just curious. If the purpose of a proof is to inform and persuade, why don't Venn diagrams count? Is it just convention or is there a more, umm, formal reason haha. Thanks!

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I'm not sure, what kind of statement could you prove using ven diagrams? –  Karolis Juodelė Feb 14 '13 at 18:18
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Maybe he's drawing $(A\cap B)^c$ as a diagram (or sequence of diagrams), and $A^c\cup B^c$ as a diagram and saying "they're equal!". –  rschwieb Feb 14 '13 at 18:33
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The utility of Venn diagrams evaporates when you look at four sets rather than just three. –  Grumpy Parsnip Feb 14 '13 at 19:55
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Not if you draw them like this: en.wikipedia.org/wiki/… –  Joe Z. Feb 14 '13 at 21:04
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@JimConant Not for five. en.wikipedia.org/wiki/File:Symmetrical_5-set_Venn_diagram.svg –  Joe Z. Feb 15 '13 at 0:37

8 Answers 8

up vote 35 down vote accepted

I don't know exactly what you have written, but I would venture to say that anything you "prove" with Venn diagrams probably has an extremely direct translation into set theory, which would certainly be an acceptable form of proof.

The strongest reason to not let you just use a Venn diagram alone is that your teacher probably wants you to verbalize your explanation. This is a key part of mathematics. Drawing a picture can really help illustrate the idea involved, but it does not always explain the connection to the logic you are working within.

There is also a huge drawback to proving things by Venn diagram: your visual preconceptions may fool you into making a mistake. This cannot happen (or happens to a much smaller degree) when you work in the language of set theory.

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Using the default Venn diagram with two intersecting circles, it is "evident" that $A\cup B\ne A\cap B$. But of course this statement is not true in general.

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It is evident only if one doesn’t understand Venn diagrams. Any tool can be misused. –  Brian M. Scott Feb 14 '13 at 22:55
    
@Brian: I have yet to meet the person who understood Venn diagrams, and had a problem with writing formal proofs. –  Asaf Karagila Feb 15 '13 at 0:47
    
@Asaf: I’ve known some who understood Venn diagrams and didn’t see any real need to say more, though they were capable of doing so. –  Brian M. Scott Feb 15 '13 at 0:49

The point is, you need to take care to be specific about what is being talked about. Symbols tend to be more specific, but mathematicians still sometimes make equivocation errors in the language. Really it boils down to not confusing the subjects and objects in a proof. As some other people have said, a Venn diagram typically applies to one set or another, so to generalize is a risky business. The same errors can occur in symbolic math manipulation, though. So there is nothing inherently invalid about the type of proof.

Venn diagrams are hardly distinct from proofs in Geometry by drawing figures.

I do agree with the comment that we are just talking about convention. Over the centuries mathematicians have changed their opinions on what constitutes a valid proof. Some of Euler's work, for example, comes to mind. It was rather arbitrary for us today to call his proofs weaker than ours today. Its a cultural chance in mindset, really.

Venn diagrams, geometrical figures... these are just concepts illustrated on paper. Thats it. How is that different than algebraic symbols written on paper? Or English? Its a language that conveys a concept. To de-emphasize the worth of one simply because it takes a different form is at best arbitrary and subjective. At worst its irrational or bigoted. Let me ask you, is there a formal proof as to why figures and shapes and Venn diagrams are less effective than symbols? It would seem to me that this would require a proof of its own.

Until such time that this is proven, the only criticism I can give the use of any one proof method over another is that we are fallible humans that may not interpret or capture the meaning of the question adequately enough in one system of proof, but we do in another. Its a failure on our own part.

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Mh I think Venn-diagrams are very helpful and in fact are the proof on several set identities, I even know mathematicans which accept them as a proof.

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I think you meant to say "I even know analysts". –  Git Gud Feb 14 '13 at 18:23

Using Venn diagrams we can prove set identities with three set variables. Why this is correct?

There is theorem that claim that Boolean algebra $\mathcal{B}$ is free on $X$ for class of Boolean algebras if for all diffrent $x_1,\dots,x_n$ it's true $x_1^{\alpha_1} \wedge x_2^{\alpha_2} \wedge \dots \wedge x_n^{\alpha_n} \neq 0$, where $$x^{\alpha}=\begin{cases}x, ~~~\alpha=1\\ x', ~~\alpha=0\end{cases}$$ So, if $A,B,C \subseteq K$ like on diagram $(V)$,

Picture

then clearly $A^{\alpha}\cap B^{\beta} \cap C^{\gamma} \neq \emptyset,$ where $$A^{\alpha}=\begin{cases}A, ~~~~~~~~~~~~~~~~~\alpha=1\\ A^c=K\backslash A, ~~\alpha=0\end{cases}$$ Let $\mathbf{\Omega}=\langle A,B,C \rangle$ be set algebra define with $\cap,\cup,^c$. Using theorem that I mentioned, we see that $\mathbf{\Omega}$ is free algebra on $X=\{A,B,C\}$ for class of Boolean algebras. Now, using theorem 1., every boolean identity, and therefore every set identity with three variables, it's true in Boolean algebras, and therefore in $\mathscr{P}(X)$.

Theorem 1. Let $\mathcal{A}$ is free algebra on $X$, $|X|=n,$ for every class algebras $\mathfrak {M}$ algebraic language $L$. If $u(x_1,\dots,x_n)=v(x_1,\dots,x_n)$ is algebraic identity in language $L$ and $u=v$ is true in $\mathcal{A}$, then $u=v$ is also true for all $\mathcal{C} \in \mathfrak{M}.$

But, four circle $A,B,C,D$ can divide plane in fourteen parts (but we want sixteen). So, $A^{\alpha}\cap B^{\beta} \cap C^{\gamma} \cap D^{\delta}= \emptyset$ for some choise of $\alpha, \beta, \gamma, \delta.$ So, for every Venn diagrams $V(A,B,C,D)$ it exist set identity $u=v$ that is true in $V(A,B,C,D)$, but which is not correct for some sets.

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The equivalent with four circles is an Euler diagram, but not a Venn Diagram. Venn diagrams can keep this going arbitrarily far, though the symbolic approach becomes much more reasonable as the diagrams quickly become unwieldy. –  Robert Mastragostino Feb 14 '13 at 19:52
    
en.wikipedia.org/wiki/… –  Martin Velez Feb 15 '13 at 1:02

The purpose of a proof is not to inform or persuade (you might not even have anybody to show your proof to). The purpose of a proof is to prove: in a sequence of small and easily verified, or known to be true, steps to prove some conclusion.

Ideally, a proof should be verifiable by a computer. You may contrast this with the fact that it is theoretically impossible to write a computer program that would verify if a given statement is provable.


Update: of course this is somewhat circular as a definition, but you cannot really define a proof. (You can of course use a metalanguage to define a proof in some other formal language, etc.) Defining a proof as a persuasion is just proposing a not completely equivalent plain-language synonym as a "definition".

To address the Venn diagrams, the problem with them is that they are usually hard to convert into a sequence of small easily verifiable steps: either you look at it and guess how to prove, or not, but a Venn diagram itself is only an illustration of a proof.

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Hmm, I don't know about the first paragraph. I think "proof as persuasion" (even if there is nobody else, and it is just persuading yourself of the truth of something) is a better description than the circular "purpose of proof is to prove." Ultimately a proof is a communication of evidence, not just a formal sequence of steps. –  rschwieb Jan 27 at 20:36
    
I think it is circular because it does not need a definition. Every dictionary is eventually circular. –  Alexey Jan 27 at 20:46
    
The problem with circularity is that it doesn't explain anything. It's probably better to have a heuristic description as "persuasion," since then you have more than one log in your raft. –  rschwieb Jan 27 at 20:55
    
I tried in other ways, but otherwise you probably just need to learn from examples. Proving that a definition of a proof defines it can be hard :). –  Alexey Jan 27 at 20:57
    
Oh I totally agree that a real definition of "proof" would be nontrivial. I forgot to mention earlier that this solution will probably not fare well until it actually addresses the point about Venn diagrams. –  rschwieb Jan 27 at 20:59

Generally because Venn diagrams only ilustrate determined examples, and most proofs you find in elementary set theory are about statements for any set, so if want to use Venn diagrams to proof something that should work for every set, you would have to actually use them for every possible set. If there are infinite sets, you're screwed.

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It depends on teacher. If I was one and could be sure that student understands Venn diagrams and has given all steps in constructing both Venn diagrams, I would agree to it.

At the start of course. Later students must learn more formal style of proof, because Venn diagrams is only for very specific problem. Although they can still use Venn diagrams as visual help in constructing proofs if they want. It's actually good and trains intuitions of the set theory.

[Note - I'm not teacher or professor.]

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