[V01] Basic notation

Module: Venn diagrams

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Genius means little more than the faculty of perceiving in an unhabitual way.

- William James

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§1. A class is defined by its members

Let us start with the concept of a class. A class or a set is simply a collection of objects. These objects are called members of the set. A class is defined by its members. So for example, we might define a class C as the class of black hats. In that case, every black hat in the world is a member of C, and anything that is not a black hat is not a member of C. If something is not a member of a class, we can also say that the object is outside the class.

Note that a class can be empty. The class of men over 5 meters tall is presumably empty since nobody is that tall. The class of plane figures that are both round and square is also empty since nothing can be both round and square. A class can also be infinite, containing an infinite number of objects. The class of even number is an example. It has infinitely many members, including 2, 4, 6, 8, and so on.

§2. Classes are represented by circles

As you can see in the diagram above, the class of black hats, C, is represented by a circle. We normally use circles to represent classes in Venn diagrams, though sometimes we also use bounded regions with different shapes, such as ovals.
We can write the name of the class, e.g. "C", or "Class C", next to the circle to indicate which class it is.
The area inside the circle represents those things which are members of the class.
The area outside the circle represents those things which are not members of the class, e.g. green hats, keys, cakes, etc.
A Venn diagram is usually enclosed by a rectangular box that represents everything in the world.

§3. Use shading to indicate an empty class

Let us now consider what shading means:

To indicate that a class is empty, we shade the circle representing that class. So the diagram above means that class A is empty.

In general, shading an area means that the class represented by the area is empty. So the second diagram above represents a situation where there isn't anything which is not a member of class A.

However, even though shading indicates emptiness, a region that is not shaded does not necessarily indicate a non-empty class. As we shall see in the next tutorial, we use a tick to indicate existence. So in the second diagram above, the circle marked A is not shaded. This does not imply that there are things which exist which are members of A. If the area is blank, this means that we do not have any information as to whether there is anything there.

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