[V06] Existence 1

We have seen how to use shading to indicate that there is nothing in the class represented by the shaded region. We now see how to use ticks to indicate existence. The basic idea is that when a tick is present in a region, it indicates that there is something in the class represented by the region. So for example, in the diagram below, we have a tick outside the circles. Since the area outside the circle represents the class of things that are neither A, nor B, nor C, the diagram is saying that something exists that is neither A nor B nor C :

There are two impotant points to remember :

1. A tick in a region says that there is something in the class represented by the region. It does not say how many things there are in that class. There might be just one, or perhaps there are many.
2. A region without a tick does not represent an empty class. Without a tick, a blank region provides no information as to whether anything exists in the class it represents. Only when a region is shaded can we say that it represents an empty class.

What about the following diagram? What does it represent?

The diagram above does NOT say "something is A". Actually it says something more specific, namely that "something is A but is not B and not C". If you have given the wrong answer, you might be thinking that the tick indicates that there is something in the class represented by the A circle. But here we use a tick to indicate existence in the class represented by the smallest bounded region that encloses the tick. In the top diagram of this page the smallest bounded area that encloses the tick is the area outside the three circles. In the diagram above, although circle A does enclose the tick, it is not the smallest bounded area that does that. That smallest region is the colored one in this diagram :

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Now see if you can determine what what these diagrams indicate.

Notice in the last diagram above, the two ticks indicate that there are two different things. What if you just want to say "Something is C but not A"? The way to do this is to put a tick across two bounded regions, as follows:

The interpretation of this digram employs the same rule as before. What the tick indicates is that there is something in the smallest closed region (the colored area) that encloses the tick. Of course, the bigger C circle also completely encloses the tick, but it is not the smallest bounded region that does that. So the tick does not mean that "something is C".

Notice that the tick does not tell us whether there is anything that is B, because it is not completely enclosed by the B circle.

See if you can explain what these diagrams represent: