Subsection 1.1.1 The notion of a set
The term set is intuitively understood by most people to mean a collection of objects that are called elements (of the set). This concept is the starting point on which we will build more complex ideas, much as in geometry where the concepts of point and line are left undefined. Because a set is such a simple notion, you may be surprised to learn that it is one of the most difficult concepts for mathematicians to define to their own liking. For example, the description above is not a proper definition because it requires the definition of a collection. (How would you define βcollectionβ?) Even deeper problems arise when you consider the possibility that a set could contain itself. Although these problems are of real concern to some mathematicians, they will not be of any concern to us. Our first concern will be how to describe a set; that is, how do we most conveniently describe a set and the elements that are in it? If we are going to discuss a set for any length of time, we usually give it a name in the form of a capital letter (or occasionally some other symbol). In discussing set
if
is an element of
then we will write
On the other hand, if
is not an element of
we write
The most convenient way of describing the elements of a set will vary depending on the specific set.
Enumeration. When the elements of a set are enumerated (or listed) it is traditional to enclose them in braces. For example, the set of binary digits is
and the set of decimal digits is
The choice of a name for these sets would be arbitrary; but it would be βlogicalβ to call them
and
respectively. The choice of a set name is much like the choice of an identifier name in programming.
Some large sets can be enumerated without actually listing all the elements. For example, the letters of the alphabet and the integers from 1 to 100 could be described as
and
The three consecutive βdotsβ are called an ellipsis. We use them when it is clear what elements are included but not listed. An ellipsis is used in two other situations. To enumerate the positive integers, we would write
indicating that the list goes on infinitely. If we want to list a more general set such as the integers between 1 and
where
is some undetermined positive integer, we might write
Standard Symbols. Sets that are frequently encountered are usually given symbols that are reserved for them alone. For example, since we will be referring to the positive integers throughout this book, we will use the symbol
instead of writing
A few of the other sets of numbers that we will use are:
the natural numbers,
the integers,
the rational numbers
the real numbers
the complex numbers
Set-Builder Notation. Another way of describing sets is to use set-builder notation. For example, we could define the rational numbers as
Note that in the set-builder description for the rational numbers:
indicates that a typical element of the set is a βfraction.β
The vertical line, is read βsuch thatβ or βwhere,β and is used interchangeably with a colon.
is an abbreviated way of saying and are integers.
Commas in mathematics are read as βand.β
The important fact to keep in mind in set notation, or in any mathematical notation, is that it is meant to be a help, not a hindrance. We hope that notation will assist us in a more complete understanding of the collection of objects under consideration and will enable us to describe it in a concise manner. However, brevity of notation is not the aim of sets. If you prefer to write
and
instead of
you should do so. Also, there are frequently many different, and equally good, ways of describing sets. For example,
and
both describe the solution set
A proper definition of the real numbers is beyond the scope of this text. It is sufficient to think of the real numbers as the set of points on a number line. The complex numbers can be defined using set-builder notation as
where
In the following definition we will leave the word βfiniteβ undefined.
Definition 1.1.1. Finite Set.
A set is a finite set if it has a finite number of elements. Any set that is not finite is an infinite set.
Definition 1.1.2. Cardinality.
Let
be a finite set. The number of different elements in
is called its cardinality. The cardinality of a finite set
is denoted
As we will see later, there are different infinite cardinalities. We canβt make this distinction until Chapter 7, so we will restrict cardinality to finite sets for now.
Subsection 1.1.2 Subsets
Definition 1.1.3. Subset.
Let
and
be sets. We say that
is a subset of
if and only if every element of
is an element of
We write
to denote the fact that
is a subset of
Example 1.1.4. Some Subsets.
If and then
If and then and
Definition 1.1.5. Set Equality.
Let
and
be sets. We say that
is equal to
(notation
) if and only if every element of
is an element of
and conversely every element of
is an element of
that is,
and
Example 1.1.6. Examples illustrating set equality.
In
Example 1.1.4,
Note that the ordering of the elements is unimportant.
The number of times that an element appears in an enumeration doesnβt affect a set. For example, if and then Warning to readers of other texts: Some books introduce the concept of a multiset, in which the number of occurrences of an element matters.
A few comments are in order about the expression βif and only ifβ as used in our definitions. This expression means βis equivalent to saying,β or more exactly, that the word (or concept) being defined can at any time be replaced by the defining expression. Conversely, the expression that defines the word (or concept) can be replaced by the word.
Occasionally there is need to discuss the set that contains no elements, namely the empty set, which is denoted by
. This set is also called the null set. Another acceptable way to denote the empty set is
It is clear, we hope, from the definition of a subset, that given any set
we have
and
If
is nonempty, then
is called an
improper subset of
All other subsets of
including the empty set, are called
proper subsets of
The empty set is an improper subset of itself.