Next: 2. Some Area Calculations Up: Math 111 Calculus I Previous: Exercises and Entertainments   Index

1. Some Notation for Sets

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 example

There are a few sets that occur very often in mathematics, and that have special names:

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

means that is an element of

and

means that is not an element of
Thus in the examples above

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 of 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

You can show that two sets are not equal, by finding an element in one of the sets that is not in the other.

In the examples above, and For every set we have

Also

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.

1.1   Definition (Box, width, height, area.) Let be real numbers with and . We define the set by

A set of this type will be called a box. If , then we will refer to the number as the width of , and we refer to as the height of .

The area of the box is the number

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''.

1.2   Definition (Unions and Intersections.) Let be a set of sets. The union of the sets is defined to be the set of all points that belong to at least one of the sets . This union is denoted by

or by
 (1.3)

The intersection of the sets is defined to be the set of points that are in every one of the sets . This intersection is denoted by

or by
 (1.4)

The index in equations 1.3 and 1.4 is called a dummy index and it can be replaced by any symbol that does not have a meaning assigned to it. Thus,

but expressions such as or will be considered to be ungrammatical.

1.5   Example. For let . Then is represented in the figure, and . Also .

In the figure below,

1.6   Definition (Set difference.) If and are sets then the set difference is the set of all points that are in but not in .

In the figure, the shaded region represents .

1.7   Exercise. A Explain why it is not true that

I will often use set relations such as

or

without explanation or justification. The second statement says that consists of the points in which are not in together with the points in that are in , and I take this and similar statements to be clear.

1.8   Definition (Intervals.) Let be real numbers with . We define the following subsets of :

A subset of is called an interval if it is equal to a set of one of these nine types. Note that and , so the empty set and a set consisting of just one point are both intervals.

1.9   Definition (End points: open and closed intervals.) If is a non-empty interval of one of the first four types in the above list, then we will say that the end points of are the numbers and . If is an interval of one of the next four types, then has the unique end point . The empty set and the interval have no end points. An interval is closed if it contains all of its end points, and it is open if it contains none of its end points.

1.10   Exercise. A Let be real numbers with . For each of the nine types of interval described in definition 1.8, decide whether an interval of the type is open or closed. (Note that some types are both open and closed, and some types are neither open nor closed.) Is the interval open? Is it closed? What about the interval ?

1.11   Exercise. In the figure below, , , and are boxes.

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 .

1.12   Exercise. Let be the set of points in such that and . Let be the set of points in such that

Make sketches of the sets and .

1.13   Exercise. Describe the sets and below in terms of unions or intersections or differences of boxes.

Next: 2. Some Area Calculations Up: Math 111 Calculus I Previous: Exercises and Entertainments   Index
Ray Mayer 2007-09-07