A set is any collection of objects. Usually
the objects we consider are things like numbers, points in the plane,
geometrical figures, or functions. Sets are often described
by listing the objects they contain inside curly braces, for
There are a few sets that occur very often in mathematics, and that have
A rational number is a number that can be expressed as a quotient of two integers. Thus a real number is rational if and only if there exist integers and with such that .
The terms ``point in the plane'' and ``ordered pair of real numbers'' are taken to be synonymous. I assume that you are familiar with the usual representation of points in the plane by pairs of numbers, and the usual way of representing geometrical objects by equations and inequalities.
Thus the set of points such that is represented in figure a, and the set of points such that and is represented in figure b. The arrowheads in figure a indicate that only part of the figure has been drawn.
The objects in a set are called elements of or points in . If is an object and is a set then
To see that observe that has exactly four elements, and none of these elements is
Let and be sets. We say that is a subset
and write if and only if every element in
is also in Two sets are considered to be equal if and
only if they have exactly the same elements. Thus
In the examples above, and For every set
The idea of set was introduced into mathematics by Georg Cantor near
the end of the nineteenth century. Since then it has become one of the
most important ideas in mathematics. In these notes we use
very little from the theory of sets, but the language of
sets will be very evident.
Remark: Notice that in the definition of box, the inequalities are ``'' and not ``''. The choice of which sort of inequality to use is somewhat arbitrary, but some of the assertions we will be making about boxes would turn out to be false if the boxes did not contain their boundaries.
In Euclid's geometry no distinction is made between sets that contain their boundaries and sets that do not. In fact the early Greek geometers did not think in terms of sets at all. Aristotle maintained that
A line cannot be made up of points, seeing that a line is a continuous thing, and a point is indivisible[25, page 123].The notion that geometric figures are sets of points is a very modern one. Also the idea that area is a number has no counterpart in Euclid's geometry, and in fact Euclid does not talk about area at all. He makes statements like
Triangles which are on equal bases and in the same parallels are equal to one another[17, vol I page 333].We interpret ``are equal to one another'' to mean ``have equal areas'', but Euclid does not define ``equal'' or mention ``area''.
The index in equations 1.3 and 1.4 is called a
dummy index and it can be replaced by any symbol that
have a meaning assigned to it. Thus,
In the figure below,
In the figure, the shaded region represents .
I will often use set relations such as
a) Express each of , , in the form .
b) Express , , and as intersections or unions or set differences of boxes. The dotted edge of indicates that the edge is missing from the set.
c) Find a box that contains .