# [V02] Everything and nothing

## §1. Intersecting circles

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?

## §2. Everything and nothing

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?