Skip to main content

Beginning Activity Beginning Activity 1: Sets and Set Notation
A4US

The theory of sets is fundamental to mathematics in the sense that many areas of mathematics use set theory and its language and notation. This language and notation must be understood if we are to communicate effectively in mathematics. At this point, we will give a very brief introduction to some of the terminology used in set theory.

A set is a well-defined collection of objects that can be thought of as a single entity itself. For example, we can think of the set of integers that are greater than 4. Even though we cannot write down all the integers that are in this set, it is still a perfectly well-defined set. This means that if we are given a specific integer, we can tell whether or not it is in the set of integers greater than 4.

The most basic way of specifying the elements of a set is to list the elements of that set. This works well when the set contains only a small number of objects. The usual practice is to list these elements between braces. For example, if the set \(C\) consists of the integer solutions of the equation \(x^2 = 9\text{,}\) we would write

\begin{equation*} C = \left\{ { - 3,3} \right\}\text{.} \end{equation*}

For larger sets, it is sometimes inconvenient to list all of the elements of the set. In this case, we often list several of them and then write a series of three dotsĀ  (\(\ldots\)) to indicate that the pattern continues. For example,

\begin{equation*} D = \left\{ 1, 3, 5, 7, \ldots, 49 \right\} \end{equation*}

is the set of all odd natural numbers from 1 to 49, inclusive.

For some sets, it is not possible to list all of the elements of a set; we then list several of the elements in the set and again use a series of three dotsĀ  (\(\ldots\)) to indicate that the pattern continues. For example, if \(F\) is the set of all even natural numbers, we could write

\begin{equation*} F = \left\{ {2,4,6, \ldots } \right\}\text{.} \end{equation*}

We can also use the three dots before listing specific elements to indicate the pattern prior to those elements. For example, if \(E\) is the set of all even integers, we could write

\begin{equation*} E = \left\{ {\ldots -6, -4, -2, 0, 2,4,6, \ldots } \right\}\text{.} \end{equation*}

Listing the elements of a set inside braces is called the roster method of specifying the elements of the set. We will learn other ways of specifying the elements of a set later in this section.

1.

Use the roster method to specify the elements of each of the following sets:

(a)

The set of real numbers that are solutions of the equation \(x^2 - 5x = 0\text{.}\)

(b)

The set of natural numbers that are less than or equal to 10.

(c)

The set of integers that are greater than \(-2\text{.}\)

2.

Each of the following sets is defined using the roster method. For each set, determine four elements of the set other than the ones listed using the roster method.

(a)

\(A = \{1, 4, 7, 10, \ldots \}\)

(b)

\(B = \{2, 4, 8, 16, \ldots \}\)

(c)

\(C = \{ \ldots, -8, -6, -4, -2, 0 \}\)

(d)

\(D = \{\ldots, -9, -6, -3, 0, 3, 6, 9, \ldots \}\)