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General settings

A set is a collection of objects, given either by a complete list of the objects in the set or by a ``defining property''. The objects in a set are aclled the elements of the set. Generally sets are denoted by capital letters $ A$, $ B$, $ E$, $ M$,$ \dots$ and their elements by lowercase letters $ a$, $ b$, $ x$, $ \dots$ .

Example 3.1.1       
(i)
$ E=\{ a ,2 , \% , * \}$ means that $ E$ is the set whose elements are the letter $ a$, the digit $ 2$, the symbols $ \%$ and the asterisk $ *$.
(ii)
$ F=\{ x \vert x$    is a positive even number $ \}$ is the set of all even natural numbers. We could have defined the set $ F$ using the following notation $ F=\{ 2,4,6,8, \dots \}$, but it is impossible to write down a complete list.

There exist sets which can be determined in both ways, i.e. by a complete list of elements and by a characteristic property, and there exist sets which can be given by only one of the ways described above. For example, the set $ \mathbb{N}$ of all the natural numbers cannot be given by a complete list, as this set is infinite (at this stage, this is an intuitive notion; we will give a precise meaning to this in chapter 7). On the other hand the set $ E=\{ a ,2 , \% , * \}$ cannot be determined by a property specific to the elements of $ E$. Let us note that the fact that the list is long is not considered as a situation where listing is impossible; giving a list of all the inhabitants of a given town is long and unilluminating, but it's possible, despite the large number of inhabitants (after all, it looks like what is done for publishing the yearly phone book of this town).

Remark 3.1.2       
  1. The order of the elements in the list is irrelevant. The notations $ \{ a,b,c,d \}$ and $ \{ b,a,d,c \}$ denote the same set.
  2. If an element is listed more than once, it counts for one element only. For example, $ \{ a,a,a,a,b,c,d \}$ is the same set s $ \{ a,b,c,d \}$.

Definition 3.1.3   There exists one set which contains no element.This set is called the empty set and is denoted by $ \emptyset$.

The empty set can appear in various ways. Let us see two examples:

Example 3.1.4       
(i)
The domain of definition of the expression $ \sqrt{x-3} +
\sqrt{1-x}$ is empty, as $ x$ has to be not less than 3 and not more than 1 at the same time.
(ii)
The set of real solutions of a quadratic equation with real coefficients is empty whenever the discriminant $ \Delta$ is negative.
(iii)
The set of sixteen feet high gnus is empty.

It is often convenient to represent sets by the so-called Venn diagrams: every point inside a closed simple curve are considered as representing an element of the set. If the set is finite (once again, we use the intuitive notion; the precise definition is given in chapter 7) we can represent them explicitely in the diagram.

Figure 1: Venn diagrams
\begin{figure}
\centering
\mbox{\subfigure[]{\epsfig{file=venn-1.eps,height=4...
... \qquad
\subfigure[]{\epsfig{file=venn-2.eps,height=4cm}}
}
\end{figure}


next up previous contents
Next: Operations with sets Up: Sets Previous: Sets   Contents
root 2002-06-10