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.

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.

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.

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 different $x_1,\dots,x_n$ 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 $(V)$,

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 choice 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.

Generally because Venn diagrams only illustrate 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 prove 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.

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.

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.]

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.

Here is another possible reason for not accepting a Venn diagram as a proof. (Here I am using the term "Venn diagram" as opposed to "Euler diagram".) If what is to be proven is a set equality, the definition of equality must be met. What is missing is a logical connection between this equality and the use of Venn diagrams. The result "If two Venn diagrams give the same picture, then the sets they refer to are equal" would suffice. So if you haven't proved this or are not allowed to use it, then you would not be allowed to use a Venn diagram to prove your set equality.

A Venn Diagram could be a proof if it captures all possible cases. In the case above where the author uses A U B does not equal A intersect B is not captured by the standard two circles with an intersection, the proof with Venn Diagrams using only one case (Two circles with a football intersection) isn't complete. If I want to prove a statement for ANY sets, I must include strange sets like the Cantor set and weird possibilities. With Venn Diagrams, you will have to prove that you have caught EVERY SINGLE possibility. This may not be feasible. Thus, we move to rigorous proof using first principles straight from the definitions and theorems. Those who said even mathematicians accept Venn Diagrams as proofs, well, I am a mathematician. There may be a couple reasons. Maybe the instructor is happy that their students were able to think things through with Venn Diagrams. Proof writing is hard and the first step is believing what you are trying to prove-which is what Venn Diagrams are for. I (mathematician) would not accept it because of the false message it sends as seen on these threads. Students may internalize that Venn diagrams are proofs for all situations.