opencourseware on critical thinking, logic, and creativity
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.
Let us now consider what shading means:
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:
|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.
Charles Franklin Kettering