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[Tutorial V01] Notation: shading

§ V01.1 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 apples. In that case, every apple in the world is a member of C, and anything that is not an apple 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.

Let us now see how Venn diagrams are used to represent classes.

§ V01.2 Classes are represented by circles

  • As you can see in the diagram above, the class of apples, 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.
  • A Venn diagram is usually enclosed by a rectangular box that represents everything in the world.

§ V01.3 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 on the left 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 on the left 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. In the second diagram on the left, the region 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.

§ V01.4 Two circles

Now let us consider a slightly more complicated diagram where we have two intersecting circles. The one on the left represents a class A. The other one represents a classs B.

Let us label the different bounded regions. Then:

  • Region 1 represents objects which are neither A nor B. This is because this area is outside both the A circle and the B circle.
  • Region 2 represents objects which have property A but which do not have property B.
  • Region 3 represents objects which have both property A and property B.
  • Region 4 represents objects which have property B but not A.

So for example, suppose B is the class of sweet things. In that case what does region three represent?
The class of sweet apples!

Furthermore, which region represents the class that contains sour lemons?
Region #1

§ V01.5 "Every" and "Nothing"

Continuing with our diagram, suppose we now shade region 2. This means that the class of things which are A but not B is empty. In other words, 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?
Every A is B. Because the A circle is inside the B circle, every member of A is also a member of B. But there might be things which are B but not A.
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Module:
Venn diagrams

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He who has imagination without learning has wings and no feet.

Joseph Joubert

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