** Module: Venn diagrams**

- V00. Introduction
- V01. Basic notation
- V02. Everything and nothing
- V03. Exercises
- V04. Three circles
- V05. Exercises
- V06. Existence 1
- V07. Existence 2
- V08. Syllogism
- V09. Exercises
- V10. Limitations of Venn diagrams

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The aim of education should be to teach us rather how to think, than what to think -- rather to improve our minds, so as to enable us to think for ourselves, than to load the memory with thoughts of other men.

- Bill Beattie

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Now let us consider a slightly more complicated diagram where we have two *intersecting* circles. The left circle represents class A. The right one represents class B.

Let us label the different bounded regions:

- Region 1 represents objects which belong to class A but not to B.
- Region 2 represents objects which belong to both A and B.
- Region 3 represents objects which belong to B but not A.
- Region 4, the area outside the two circles, represents objects that belong to neither A nor B.

So for example, suppose A is the class of apples, and B is the class of sweet things. In that case what does region 2 represent?

Furthermore, which region represents the class that contains sour lemons that are not sweet?

Continuing with our diagram, suppose we now shade region 1. This means that the class of things which belong to A but not B is empty. Or more simply, **every A is a B**. ( It might be useful to note that this is equivalent to saying that if anything is an A, it is also a B. ) This is an important point to remember. Whenever you want to represent "every A is B", shade the area within the A circle that is outside the B circle.

What if we shade the middle region where A and B overlaps? This is the region representing things which are both A and B. So shading indicates that **nothing is both A and B**. If you think about it carefully, you will see that "Nothing is both A and B" says the same thing as "No A is a B" and "No B is an A". Make sure that you understand why these claims are logically equivalent!

Incidentally, we could have represented the same information by using two non-overlapping circles instead.

What about the diagram on the left? What do you think it represents?