Critical thinking web

[Tutorial V08] Limitations

Although Venn diagrams can help us reason about classes of objects, they also have many limitations. First of all, the diagrams can become too complicated to deal with if we are reasoning about many classes of objects. So far in our tutorials we have considered Venn diagrams with at most three circles. It is possible to add more bounded regions if we are dealing with more than three classes, but then the resulting diagrams will become very difficult to handle. For example, if we are reasoning about four different classes of objects, we shall have to make use of a diagram like the one above. As you can see, interpretation is more difficult than before. It is also easier to make mistakes when we encode information in such diagrams.

The other problem with Venn diagrams is that they have limited expressive power. What this means is that there are many pieces of information that cannot be accurately represented. For example, our system of notation allows us to talk about classes of objects, but not particular individual objects. For example, to say that a and b are cats and c and d are not, we might have to introduce new symbols, using dots to represent individuals, as in the diagram below:

However, even with this new notation, there are still other pieces of information that cannot be represented, such as :

  • Either Felix is a cat or it is a dog.
  • If Peter is taller than Mary then Peter is older than Mary.

Perhaps it might be possible to introduce additional new symbols to represent such ideas. But then the system of Venn diagrams will get really complicated and difficult to use. So now that we know the limitations of Venn diagrams, we should be in a better position to know when they are useful and when they are not.

§ V08.1 Exercises

Question 1.

See if you can invent a new notation for representing a sentence such as "Felix is either a cat or a dog.".
Maybe use a line to connect two dots to indicate that an object is one of the connected ones?

Question 2.

Compare the following diagram with the one at the top of this page. Is the diagram below a suitable one for showing the relationships between four sets of objects?

There is no region where only the yellow and blue, or only the pink and green circles meet.

Question 3.

What about the image above? Which area corresponds to those items which are A and D, but neither B nor C?
It is the purple square at the lower left corner.

Question 4.

This is a difficult question. How would you draw a Venn diagram to represent the relationships between five sets of objects?

You can find one way to draw the diagram on Wikipedia here.

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