4.1 Set Notation

Sets and Ways to Represent Them

Think back to your kitchen organization. If the drawer is the set, then the forks and knives are elements in the set. Sets can be described in a number of different ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and by Venn diagrams. Sets are typically designated with capital letters. The simplest way to represent a set with only a few members is the roster (or listing) method, in which the elements in a set are listed, enclosed by curly braces and separated by commas. For example, if $F$ represents the set of utensils in our kitchen drawer, we can represent $F$ by using the following set notation with the roster method:

$$F=\left\{\mathrm{fork,\; spoon,\; knife,\; meat\; thermometer,\; can\; opener}\right\}$$

All the sets we have considered so far have been well-defined sets. A well-defined set clearly communicates whether an element is a member of the set or not. A set of your three favorite sports is well-defined once you state the three sports you have chosen. Two people can choose different sets without making either fixed list ambiguous. By contrast, the phrase “the top 10 songs of 2021” needs a stated ranking rule or an explicitly chosen list before membership is clear. The set of letters in a stated name is also well-defined because the letters can be identified. The NFL wide receiver, Chad Johnson, famously changed his name to Chad Ochocinco to match his jersey number of 85.

On January 20, 2021, Kamala Harris was sworn in as the first woman vice president of the United States of America. If we were to consider the set of all women vice presidents of the United States of America prior to January 20, 2021, this set would be known as an empty set; the number of people in this set is 0, since there were no women vice presidents before Harris. The empty set, also called the null set, is written symbolically using a pair of braces, $\left\{\right\}$, or a zero with a slash through it, $\varnothing$.

For larger sets that have a natural ordering, sometimes an ellipsis is used to indicate that the pattern continues. It is common practice to list the first three elements of a set to establish a pattern, write the ellipsis, and then provide the last element. Consider the set of all lowercase letters of the English alphabet, $A$. This set can be written symbolically as $A=\{\mathit{a,\; b,\; c,}\dots \mathit{,\; z}\}$.

The sets we have been discussing so far are finite sets. They all have a limited or fixed number of elements. We also use an ellipsis for infinite sets, which have an unlimited number of elements, to indicate that the pattern continues. For example, in set theory, the set of natural numbers, which is the set of all positive counting numbers, is represented as $\mathrm{\mathbb{N}}=\{1,2,3,\dots \}$.

Notice that for this set, there is no element following the ellipsis. This is because there is no largest natural number; you can always add one more to get to the next natural number. Because the set of natural numbers grows without bound, it is an infinite set.

Our number system is made up of several different infinite sets of numbers. The set of integers, $\mathrm{\mathbb{Z}},$ is another infinite set of numbers. It includes all the positive and negative counting numbers and the number zero. There is no largest or smallest integer.

A shorthand way to write sets is with the use of set builder notation, which is a verbal description or formula for the set. For example, the set of all lowercase letters of the English alphabet, $A$, written in set builder notation is:

$$A=\left\{x|x\mathrm{is\; a\; lowercase\; letter\; of\; the\; English\; alphabet.}\right\}$$

This is read as, “Set $A$ is the set of all elements $x$ such that $x$ is a lowercase letter of the English alphabet.”

Computing the Cardinal Value of a Set

Almost all the sets most people work with outside of pure mathematics are finite sets. For these sets, the cardinal value or cardinality of the set is the number of elements in the set. For finite set $A$, the cardinality is denoted symbolically as $n\left(A\right)$. For example, a set that contains four elements has a cardinality of 4.

How do we measure the cardinality of infinite sets? The smallest infinite cardinality is the cardinality of the set of natural numbers, or counting numbers, $\mathrm{\mathbb{N}}=\{1,2,3,\dots \}$. This set has a cardinality of ${\mathrm{\aleph }}_0$ (pronounced "aleph-null"). All sets that have the same cardinality as the set of natural numbers are countably infinite. This concept, as well as notation using aleph, was introduced by mathematician Georg Cantor who once said, “A set is a Many that allows itself to be thought of as a One.”

Now that we have learned to represent finite and infinite sets using both the roster method and set builder notation, we should also be able to determine if a set is finite or infinite based on its verbal or symbolic description. One way to determine if a set is finite or not is to determine the cardinality of the set. A set is finite if its cardinality is a nonnegative integer, including 0 for the empty set.