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# Why aren't Venn diagrams rigourous? watch

1. I've been told on multiple occasions that Venn diagrams are not a sufficiently rigourous representation of sets (for, for example, proving the equivalence of two expressions involving sets). However, I can't see what the problem with them is! Can anybody explain why they aren't satisfactory?
2. I'm not entirely sure, but I think it's more of a case of mathematical acceptance as an argument.
3. I suppose if you wanted to talk about an arbitary number of sets then you would end up having to lose the diagram so you might as well just stick with an argument that can encompass both.

As to rigour that is an entirely different question but some people seem to think rigour and generaitly go hand in hand so they might be the campaigners for outing diagrams (and saying they aren't rigorous is a pretty good way to out them).
4. The only issue I can really see is that it's hard to "show working" with a Venn diagram.
5. (Original post by DFranklin)
The only issue I can really see is that it's hard to "show working" with a Venn diagram.
This.
6. (Original post by marcusmerehay)
I'm not entirely sure, but I think it's more of a case of mathematical acceptance as an argument.
Surely those concerns could be allayed by proving some theorem which develops a (truth-preserving) algorithmic process for constructing a Venn diagram from the algebraic expression.
7. Typically, a Venn diagram starts with a nice rectangle, which you might possibly call 'the Universal set' inside which all the things you're interested in live. So S1 questions ask about choosing a playing card or the numbers from 1 to 10. As long as you stick to well-behaved choices like that, there's no problem.

Trouble is, you might try to define your Universal set as 'the set of all possible things'. It all starts to unravel then, as the Russell paradox shows. That's about first-year Uni Maths stuff. After you've had your fingers burned by that one, you get to be very careful.
8. Maths is only rigorous when you have an axiomatic basis for what you're saying. Venn diagrams don't have any axioms (not that I've learnt anyway) governing exactly what properties they have and how they are formed, and so cannot be rigorous.
9. (Original post by ian.slater)
Typically, a Venn diagram starts with a nice rectangle, which you might possibly call 'the Universal set' inside which all the things you're interested in live. So S1 questions ask about choosing a playing card or the numbers from 1 to 10. As long as you stick to well-behaved choices like that, there's no problem.

Trouble is, you might try to define your Universal set as 'the set of all possible things'. It all starts to unravel then, as the Russell paradox shows. That's about first-year Uni Maths stuff. After you've had your fingers burned by that one, you get to be very careful.
Why can't we just resolve that problem by considering the Venn Diagram universe to be defined by the ZF axiom of restricted comprehension?
10. (Original post by Kolya)
Why can't we just resolve that problem by considering the Venn Diagram universe to be defined by the ZF axiom of restricted comprehension?
Because no-one understands what this is...
11. It would be useful to know exactly what you had in mind when you asked the original question.

My take: in normal usage, there's no problem with using Venn diagrams. The main problem is that it's very hard to give a step-by-step solution, as opposed to "well, here's the Venn diagram which looks exactly like what I want to prove, so what I want to prove is true".

Yes, you may have problems with things like Russell's paradox, but it's not like you can't have those problems using normal set notation (after all, that is how Russell's paradox is usually expressed). If you're doing "proper" axiomatic set theory, I'd say you'll run into problems with trying to annotate your diagram (and working out how to draw a venn diagram for an uncountable collection of sets etc.) way before you hit the paradoxes. [i.e. Yeah, Venn diagrams aren't much use for axiomatic set theory, but it's not really because of Russell's paradox].
12. (Original post by DFranklin)
It would be useful to know exactly what you had in mind when you asked the original question.
Really nothing more than proving stuff like is equivalent to
13. (Original post by Kolya)
Really nothing more than proving stuff like is equivalent to
To my mind, there's no "rigour" issue, but it's going to be hard to avoid a Proof by Venn Diagram looking like "it's true because, um, it's true".

(On the other hand, Proof by Truth Table is generally seen as acceptable, and is really no different).
14. (Original post by DFranklin)
To my mind, there's no "rigour" issue, but it's going to be hard to avoid a Proof by Venn Diagram looking like "it's true because, um, it's true".

(On the other hand, Proof by Truth Table is generally seen as acceptable, and is really no different).
I suppose one of the main problems of a Venn diagram is that it's like a single row of a truth table, when you draw it you draw it with a definite level of intersection etc. To prove it works you'd have to consider it for all possible situations, which means a truth table would be easier.

So in essence, multiple Venn diagrams are fine, a single 1 just doesn't consider all the possibilities.

I also think it's dangerous as venn diagrams aren't well-defined. There are no real rules as to how they work, just conventions on ho wwe draw them.
15. My N&S supervisor accepted my venn diagrams as proofs for simple < 4 set identities.
16. (Original post by JoMo1)
I suppose one of the main problems of a Venn diagram is that it's like a single row of a truth table, when you draw it you draw it with a definite level of intersection etc. To prove it works you'd have to consider it for all possible situations, which means a truth table would be easier.
Not really. If you're only talking about 3 sets (i.e. the A n (B u C) type identities Kolya mentioned), then a single Venn diagram covers all 8 rows of the truth table.

With 4 sets life gets trickier, but then again the bigger problem is "It's not actually possible to draw a venn diagram to reproduce this situation". In other words you're much more likely to find "Venn diagrams are a really rubbish way of doing this" than "You can do this with a Venn diagram but it isn't rigorous".

It's also worth realising that "formalism for the sake of formalism" doesn't actually make things more rigorous, at least past an elementary level. It's quite interesting reading Littlewood's Mathematician's Miscellany where he says (paraphrased) "For the professional mathematician, sometimes a sketch is enough to prove a result rigorously".
17. What is rigour?

I teach my students that a tangent is at 90 degrees to the radius, because it is a ball sitting on a table.

If the table is not horizontal, the ball rolls off, so the table must be at 90 degrees to the radius.

This is at least as 'rigourous' as a proof that you can put a ball on a table and have it stay there (which we all know is true)

Surely Venn diagrams are just as rigourous as any proof in geometry that relies on a diagram.

It would depend upon the use made of the diagram rather than the fact that a diagram has been used.
18. (Original post by stevencarrwork)
Surely Venn diagrams are just as rigourous as any proof in geometry that relies on a diagram.
So not rigorous at all

Proof every triangle is isosceles: http://www.jimloy.com/geometry/every.htm

I really don't get that thing about tangents and balls either
19. (Original post by stevencarrwork)
What is rigour?

I teach my students that a tangent is at 90 degrees to the radius, because it is a ball sitting on a table.

If the table is not horizontal, the ball rolls off, so the table must be at 90 degrees to the radius.

This is at least as 'rigourous' as a proof that you can put a ball on a table and have it stay there (which we all know is true)

Surely Venn diagrams are just as rigourous as any proof in geometry that relies on a diagram.

It would depend upon the use made of the diagram rather than the fact that a diagram has been used.
You teach them that and claim rigour?

How do you know the actual angle of the table keeping it stable is a right-angle?

You have also only demonstrated correlation between a change from 90 and the ball rolling off, that does not imply causality.

I agree it's a reasonable way of getting kids to understand the concept of it, but it's not rigorous.

(Original post by DFranklin)
Not really. If you're only talking about 3 sets (i.e. the A n (B u C) type identities Kolya mentioned), then a single Venn diagram covers all 8 rows of the truth table.

With 4 sets life gets trickier, but then again the bigger problem is "It's not actually possible to draw a venn diagram to reproduce this situation". In other words you're much more likely to find "Venn diagrams are a really rubbish way of doing this" than "You can do this with a Venn diagram but it isn't rigorous".

It's also worth realising that "formalism for the sake of formalism" doesn't actually make things more rigorous, at least past an elementary level. It's quite interesting reading Littlewood's Mathematician's Miscellany where he says (paraphrased) "For the professional mathematician, sometimes a sketch is enough to prove a result rigorously".
I still don't agree that it covers all possible situations, even in the simple cases. You have ot consider what happens when things don't intersect with each other, when some things are the empty set etc.

I come back to my original point: Venn Diagrams are not well-defined, if they are then you can use them, but I've never been taught it and noone seems to have mentioned a general acceptance of how they work.
20. (Original post by JoMo1)
I still don't agree that it covers all possible situations, even in the simple cases. You have ot consider what happens when things don't intersect with each other, when some things are the empty set etc.
For a "3 set identity", all you need to do is verify it holds for elements in each of the eight subsets:

A n B n C
A n B n C^c
...
A^c n B^c n C^c

(That is, show that if , then either x is in both sides of the identity, or absent from both sides of the identity. And repeat for the other 7 cases. There's nothing unrigorous about doing this by Venn diagram (aside from the previously mentioned fact that there's no obvious working)).

I come back to my original point: Venn Diagrams are not well-defined, if they are then you can use them, but I've never been taught it and noone seems to have mentioned a general acceptance of how they work.
Lots of graphs and diagrams used in proofs aren't defined rigorously either. To a professional mathematician, that doesn't necessarily matter, as long as it's obvious what is being said.

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