## Learning Objectives

After completing this section, you should be able to:

- Represent sets in a variety of ways.
- Represent well-defined sets and the empty set with proper set notation.
- Compute the cardinal value of a set.
- Differentiate between finite and infinite sets.
- Differentiate between equal and equivalent sets.

## 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 our set of flatware, we can represent $F$ by using the following set notation with the roster method:

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

## Example 1.1

### Writing a Set Using the Roster or Listing Method

Write a set consisting of your three favorite sports and label it with a capital $S$.

### Solution

There are multiple possible answers depending on what your three favorite sports are, but any answer must list three different sports separated by commas, such as the following:

## Your Turn 1.1

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. The members of a well-defined set are fixed and do not change over time. Consider the following question. What are your top 10 songs of 2021? You could create a list of your top 10 favorite songs from 2021, but the list your friend creates will not necessarily contain the same 10 songs. So, the set of your top 10 songs of 2021 is not a well-defined set. On the other hand, the set of the letters in your name is a well-defined set because it does not vary (unless of course you change your name). The NFL wide receiver, Chad Johnson, famously changed his name to Chad Ochocinco to match his jersey number of 85.

## Example 1.2

### Identifying Well-Defined Sets

For each of the following collections, determine if it represents a well-defined set.

- The group of all past vice presidents of the United States.
- A group of old cats.

### Solution

- The group of all past vice presidents of the United States is a well-defined set, because you can clearly identify if any individual was or was not a member of that group. For example, Britney Spears is not a member of this set, but Joe Biden is a member of this set.
- A group of old cats is not a well-defined set because the word old is ambiguous. Some people might consider a seven-year-old cat to be old, while others might think a cat is not old until it is 13 years old. Because people can disagree on what is and what is not a member of this group, the set is not well-defined.

## Your Turn 1.2

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, $\left\{\phantom{\rule{0.28em}{0ex}}\right\}$, or a zero with a slash through it, $\varnothing $.

## Checkpoint

*The set containing the number* $0,\left\{0\right\}$, *is a set with one element in it. It is not the same as the empty set*, $\left\{\phantom{\rule{0.28em}{0ex}}\right\}$, *which does not have any elements in it. Symbolically:* $\left\{0\right\}\ne \left\{\phantom{\rule{0.28em}{0ex}}\right\}$*.*

## Example 1.3

### Representing the Empty Set Symbolically

Represent each of the following sets symbolically.

- The set of prime numbers less than 2.
- The set of birds that are also mammals.

### Solution

- A prime number is a natural number greater than 1 that is only divisible by one and itself. Since there are no prime numbers less than 2, this set is empty, and we can represent it symbolically as follows: $\left\{\phantom{\rule{0.28em}{0ex}}\right\}\mathrm{or}\varnothing .$ These two different symbols for the empty set can be used interchangeably.
- The set of birds and the set of mammals do not intersect, so the set of birds that are also mammals is empty, and we can represent it symbolically as $\varnothing \phantom{\rule{0.28em}{0ex}}\mathrm{or}\left\{\phantom{\rule{0.28em}{0ex}}\right\}\mathrm{.}$

## Your Turn 1.3

## Who Knew?

### The Number Zero

We use the number zero to represent the concept of nothing every day. The machine language of computers is binary, consisting only of zeros and ones, and even way before that, the number zero was a powerful invention that allowed our understanding of mathematics and science to develop. The historical record shows the Babylonians first used zeros around 300 B.C., while the Mayans developed and began using zero separately around 350 A.D. What is considered the first formal use of zero in arithmetic operations was developed by the Indian mathematician Brahmagupta around 650 A.D.

Brahmagupta, Mathematician and Astronomer

Another interesting feature of the number zero is that although it is an even number, it is the only number that is neither negative nor positive.

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.

## Example 1.4

### Writing a Finite Set Using the Roster Method and an Ellipsis

Write the set of even natural numbers including and between 2 and 100, and label it with a capital $E$. Include an ellipsis.

### Solution

Write the label, $E$, followed by an equal sign and then a bracket. Write the first three even numbers separated by commas, beginning with the number two to establish a pattern. Next, write the ellipsis followed by a comma and the last number in the list, 100. Finally, write the closing bracket to complete the set.

Write the label, $E$, followed by an equal sign and then a bracket.

Write the first three even numbers separated by commas, beginning with the number 2 to establish a pattern.

Next, write the ellipsis followed by a comma and the last number in the list, 100.

Finally, write the closing bracket to complete the set.

## Your Turn 1.4

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.

## Example 1.5

### Writing an Infinite Set Using the Roster Method and Ellipses

Write the set of integers using the roster method, and label it with a $\mathrm{\mathbb{Z}}$.

### Solution

**Step 1:** As always, we write the label and then the opening bracket. Because the negative counting numbers are infinite, to represent that the pattern continues without bound to the left, we must use an ellipsis as the first element in our list.

**Step 2:** We place a comma and follow it with at least three consecutive integers separated by commas to establish a pattern.

**Step 3:** Add an ellipsis to the end of the list to show that the set of integers continues without bound to the right.

Complete the list with a closing bracket. The set of integers may be represented as follows: $\mathrm{\mathbb{Z}}=\{\dots ,-2,-1,0,1,2,\dots \}.$

## Your Turn 1.5

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:

$A=\left\{x|x\phantom{\rule{0.28em}{0ex}}\mathrm{is\; a\; lowercase\; letter\; of\; the\; English\; alphabet.}\right\}$

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

## Example 1.6

### Writing a Set Using Set Builder Notation

Using set builder notation, write the set $B$ of all types of balls. Explain what the notation means.

### Solution

The verbal description of the set is, “Set $B$ is the set of all elements $b$ such that $b$ is a ball.” This set can be written in set builder notation as follows: $B=\left\{b|b\phantom{\rule{0.28em}{0ex}}\text{is a ball}.\right\}$

## Your Turn 1.6

## Example 1.7

### Writing Sets Using Various Methods

Consider the set of letters in the word “happy.” Determine the best way to represent this set, and then write the set using either the roster method or set builder notation, whichever is more appropriate.

### Solution

Because the letters in the word “happy” consist of a small finite set, the best way to represent this set is with the roster method. Choose a label to represent the set, such as $H$.

$H=\left\{\mathrm{h},\mathrm{a},\mathrm{p},\mathrm{y}\right\}$.

Notice that the letter “p” is only represented one time. This occurs because when representing members of a set, each unique element is only listed once no matter how many times it occurs. Duplicate elements are never repeated when representing members of a set.

## Your Turn 1.7

## 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 set is 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.”

## Example 1.8

### Computing the Cardinal Value of a Set

Write the cardinal value of each of the following sets in symbolic form.

- $F=\left\{\text{fork, spoon, knife, meat thermometer, can opener}\right\}$
- The empty set.

### Solution

- There are 5 distinct elements in set $F$: a fork, a spoon, a knife, a meat thermometer, and a can opener. Therefore, the cardinal value of set $F$ is 5 and written symbolically as $n\left(F\right)=5.$
- Because the empty set does not have any elements in it, the cardinality of the empty set is zero. Symbolically we write this as: $n(\varnothing )=0.$

## Your Turn 1.8

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. If the cardinality of a set is a natural number, then the set is finite.

## Example 1.9

### Differentiating Between Finite and Infinite Sets

Classify each of the following sets as infinite or finite.

- $E=\left\{2,4,6,8,10\right\}$
- $A$ is the set of lowercase letters of the English Alphabet, $A=\{\mathrm{a},\mathrm{b},\mathrm{c},\dots ,\mathrm{z}\}$.
- $\mathrm{\mathbb{Q}}=\left\{{\displaystyle \frac{p}{q}}|p\phantom{\rule{0.28em}{0ex}}\mathrm{and}\phantom{\rule{0.28em}{0ex}}q\phantom{\rule{0.28em}{0ex}}\mathrm{are}\phantom{\rule{0.28em}{0ex}}\mathrm{integers}\phantom{\rule{0.28em}{0ex}}\mathrm{and}\phantom{\rule{0.28em}{0ex}}q\ne 0\right\}$

### Solution

- $n\left(E\right)=5$. Since 5 is a natural number, the set is finite.
- $n\left(A\right)=26$. Since 26 is a natural number, the set is finite.
- Set $\mathrm{\mathbb{Q}}$ is the set of rational numbers or fractions. Because the set of integers is a subset of the set of rational numbers, and the set of integers is infinite, the set of rational numbers is also infinite. There is no smallest or largest rational number.

## Your Turn 1.9

## Equal versus Equivalent Sets

When speaking or writing we tend to use equal and equivalent interchangeably, but there is an important distinction between their meanings. Consider a new Ford Escape Hybrid and a new Toyota Rav4 Hybrid. Both cars are hybrid electric sport utility vehicles; in that sense, they are equivalent. They will both get you from place to place in a relatively fuel-efficient way. In this example we are comparing the single member set {Toyota Rav4 Hybrid} to the single member set {Ford Escape Hybrid}. Since these two sets have the same number of elements, they are also equivalent mathematically, meaning they have the same cardinality. But they are not equal, because the two cars have different looks and features, and probably even handle differently. Each manufacturer will emphasize the features unique to their vehicle to persuade you to buy it; if the SUVs were truly equal, there would be no reason to choose one over the other.

Now consider two Honda CR-Vs that are made with exactly the same parts, on the same assembly line within a few minutes of each other—these SUVs are equal. They are identical to each other, containing the same elements without regard to order, and the only differentiator when making a purchasing decision would be varied pricing at different dealerships. The set {Honda CR-V} is equal to the set {Honda CR-V}. Symbolically, we represent equal sets as $A=B$ and equivalent sets as $A\sim B$.

Now, let us consider a Toyota dealership that has 10 RAV4s on the lot, 8 Prii, 7 Highlanders, and 12 Camrys. There is a one-to-one relationship between the set of vehicles on the lot and the set consisting of the number of each type of vehicle on the lot. Therefore, these two sets are equivalent, but not equal. The set {RAV4, Prius, Highlander, Camry} is equivalent to the set {10, 8, 7, 12} because they have the same number of elements.

## Checkpoint

*If two sets are equal, they are also equivalent, because equal sets also have the same cardinality.*

## Example 1.10

### Differentiating Between Equivalent and Equal Sets

Determine if the following pairs of sets are equal, equivalent, or neither.

- $E=\left\{2,4,6,8,10\right\}$ and $F=\left\{\mathrm{fork,\; spoon,\; knife,\; meat\; thermometer,\; can\; opener}\right\}$
- The empty set and the set of prime numbers less than 2.
- The set of vowels in the word happiness and the set of consonants in the word happiness.

### Solution

- Sets
*E*and*F*both have a cardinal value of 5, but the elements in these sets are different. So, the two sets are equivalent, but they are not equal: $E\sim F$. - The set of prime numbers consists of the set of counting numbers greater than one that can only be divided evenly by one and itself. The set of prime numbers less than 2 is an empty set, since there are no prime numbers less than 2. Therefore, these two sets are equal (and equivalent).
- The set of vowels in the word happiness is $\left\{\mathrm{a},\phantom{\rule{0.28em}{0ex}}\mathrm{i},\phantom{\rule{0.28em}{0ex}}\mathrm{e}\right\}$ and the set of consonants in the word happiness is $\left\{\mathrm{h},\phantom{\rule{0.28em}{0ex}}\mathrm{p},\phantom{\rule{0.28em}{0ex}}\mathrm{n},\phantom{\rule{0.28em}{0ex}}\mathrm{s}\right\}.$ The cardinal value of these sets two sets is $n\left(\{\mathrm{a},\phantom{\rule{0.28em}{0ex}}\mathrm{i},\phantom{\rule{0.28em}{0ex}}\mathrm{e}\}\right)=3$ and $n\left(\{\mathrm{h},\phantom{\rule{0.28em}{0ex}}\mathrm{p},\phantom{\rule{0.28em}{0ex}}\mathrm{n},\phantom{\rule{0.28em}{0ex}}\mathrm{s}\}\right)=4,$ respectively. Because the cardinality of the two sets differs, they are not equivalent. Further, their elements are not identical, so they are also not equal.

## Your Turn 1.10

## People in Mathematics

### Georg Cantor

Georg Cantor, the father of modern set theory, was born during the year 1845 in Saint Petersburg, Russia and later moved to Germany as a youth. Besides being an accomplished mathematician, he also played the violin. Cantor received his doctoral degree in Mathematics at the age of 22.

In 1870, at the age of 25 he established the uniqueness theorem for trigonometric series. His most significant work happened between 1874 and 1884, when he established the existence of transcendental numbers (also called irrational numbers) and proved that the set of real numbers are uncountably infinite—despite the objections of his former professor Leopold Kronecker.

Cantor published his final treatise on set theory in 1897 at the age of 52, and was awarded the Sylvester Medial from the Royal Society of London in 1904 for his contributions to the field. At the heart of Cantor’s work was his goal to solve the continuum problem, which later influenced the works of David Hilbert and Ernst Zermelo.

References:

Wikipedia contributors. “Cantor.” Wikipedia, The Free Encyclopedia, 23 Mar. 2021. Web. 20 Jul. 2021.

Akihiro Kanamori, “Set Theory from Cantor to Cohen,” Editor(s): Dov M. Gabbay, Akihiro Kanamori, John Woods, *Handbook of the History of Logic*, North-Holland, Volume 6, 2012.