One purpose of Chapter 15 is to support a co-requisite approach to teaching College Algebra or Precalculus. ^[1] (Please read The New Preface for details as to why we decided to organize our book in order to support this pedagogy. Also, to us Precalculus = College Algebra + College Trigonometry without the formalization of limits. This distinction is not universally agreed upon so we felt the need to point it out.) Our goal is to provide instructors with supplemental material linked to the main textbook that can be used to support students who have minor gaps in their pre-college mathematical backgrounds. To that end, we have written a collection of somewhat independent sections designed to review the concepts, skills and vocabulary that we believe are prerequisite to a rigorous, college-level Precalculus course. This review is not designed to teach the material to students who have never seen it before so the presentation is more succinct and the exercise sets are shorter than those usually found in an Intermediate Algebra or high school Algebra II text. Some of this material (like adding fractions and plotting points) is used throughout the text but, where appropriate, we have referenced specific sections of the main body of the Precalculus text in an effort to assist faculty who would like to assign the Appendix as just in time review reading to their students.

We begin this section with the definition of a concept that is central to all of Mathematics.

For example, the collection of letters that make up the word “smolko” is well-defined and is a set, but the collection of the worst Math teachers in the world is not well-defined and therefore is not a set. ^[2] (For a more thought-provoking example, consider the collection of all things that do not contain themselves - this leads to the famous paradox known as Russell’s Paradox.)

In general, there are three ways to describe sets and those methods are listed below.

1. The Verbal Method: Use a sentence to describe the elements the set.
2. The Roster Method: Begin with a left brace ‘\(\{\)’, list each element of the set only once and then end with a right brace ‘\(\}\)’.
3. The Set-Builder Method: A combination of the verbal and roster methods using a “dummy variable” such as \(x\) and conditions on that variable.

Let \(S\) be the set described verbally as the set of letters that make up the word “smolko”. A roster description of \(S\) is \(\left \{ s, m, o, l, k \right \}\). Note that we listed ‘o’ only once, even though it appears twice in the word “smolko”. Also, the order of the elements doesn’t matter, so \(\left \{ k, l, m, o, s \right \}\) is also a roster description of \(S\). A set-builder description of \(S\) is: \(\{ x \, | \, \mbox {$x$ is a letter in the word ``smolko''}\}\). The way to read this is ‘The set of elements \(x\) such that \(x\) is a letter in the word “smolko”.’ In each of the above cases, we may use the familiar equals sign ‘\(=\)’ and write \(S = \left \{ s, m, o, l, k \right \}\) or \(S = \{ x \, | \, \mbox {$x$ is a letter in the word ``smolko''}\}\).

Notice that \(m\) is in \(S\) but many other letters, such as \(q\), are not in \(S\). We express these ideas of set inclusion and exclusion mathematically using the symbols \(m \in S\) (read ‘\(m\) is in \(S\)’) and \(q \notin S\) (read ‘\(q\) is not in \(S\)’). More precisely, we have the following.

Now let’s consider the set \(C = \{ x \, | \, \mbox {$x$ is a consonant in the word ``smolko''}\}\). A roster description of \(C\) is \(C = \{ s, m, l, k\}\). Note that by construction, every element of \(C\) is also in \(S\). We express this relationship by stating that the set \(C\) is a subset of the set \(S\), which is written in symbols as \(C \subseteq S\). The more formal definition is given at the top of the next page.

In our previous example, \(C \subseteq S\) yet not vice-versa since \(o \in S\) but \(o \notin C\). Additionally, the set of vowels \(V = \{ a, e, i, o, u\}\), while it does have an element in common with \(S\), is not a subset of \(S\). (As an added note, \(S\) is not a subset of \(V\), either.) We could, however, build a set which contains both \(S\) and \(V\) as subsets by gathering all of the elements in both \(S\) and \(V\) together into a single set, say \(U = \{ s, m, o, l, k, a, e, i, u\}\). Then \(S \subseteq U\) and \(V \subseteq U\). The set \(U\) we have built is called the union of the sets \(S\) and \(V\) and is denoted \(S \cup V\). Furthermore, \(S\) and \(V\) aren’t completely different sets since they both contain the letter ‘o.’ The intersection of two sets is the set of elements (if any) the two sets have in common. In this case, the intersection of \(S\) and \(V\) is \(\{ o\}\), written \(S \cap V = \{ o \}\). We formalize these ideas below.

The key words in Definition intersectionunion to focus on are the conjunctions: ‘intersection’ corresponds to ‘and’ meaning the elements have to be in both sets to be in the intersection, whereas ‘union’ corresponds to ‘or’ meaning the elements have to be in one set, or the other set (or both). Please note that this mathematical use of the word ‘or’ differs than how we use ‘or’ in spoken English. In Math, we use the inclusive or which allows for the element to be in both sets. At a restaurant if you’re asked “Do you want fries or a salad?” you must pick one and only one. This is known as the exclusive or and it plays a role in other Math classes. For our purposes it is good enough to say that for an element to belong to the union of two sets it must belong to at least one of them.

Returning to the sets \(C\) and \(V\) above, \(C \cup V = \{ s, m, l, k, a, e, i, o, u\}\). ^[3] (Which just so happens to be the same set as \(S \cup V\).) Their intersection, however, creates a bit of notational awkwardness since \(C\) and \(V\) have no elements in common. While we could write \(C \cap V = \{ \}\), this sort of thing happens often enough that we give the set with no elements a name.

As promised, the empty set is the set containing no elements since no matter what ‘\(x\)’ is, ‘\(x = x\).’ Like the number ‘\(0\),’ the empty set plays a vital role in mathematics. ^[4] (Sadly, the full extent of the empty set’s role will not be explored in this text.) We introduce it here more as a symbol of convenience as opposed to a contrivance ^[5] (Actually, the empty set can be used to generate numbers - mathematicians can create something from nothing!) because saying that \(C \cap V = \emptyset \) is unambiguous whereas \(\{ \}\) looks like a typographical error.

A nice way to visualize the relationships between sets and set operations is to draw a Venn Diagram. A Venn Diagram for the sets \(S\), \(C\) and \(V\) is drawn below.

In the Venn Diagram above we have three circles - one for each of the sets \(C\), \(S\) and \(V\). We visualize the area enclosed by each of these circles as the elements of each set. Here, we’ve spelled out the elements for definitiveness. Notice that the circle representing the set \(C\) is completely inside the circle representing \(S\). This is a geometric way of showing that \(C \subseteq S\). Also, notice that the circles representing \(S\) and \(V\) overlap on the letter ‘o’. This common region is how we visualize \(S \cap V\). Notice that since \(C \cap V = \emptyset \), the circles which represent \(C\) and \(V\) have no overlap whatsoever.

All of these circles lie in a rectangle labeled \(U\) for the ‘universal’ set. A universal set contains all of the elements under discussion, so it could always be taken as the union of all of the sets in question, or an even larger set. In this case, we could take \(U = S \cup V\) or \(U\) as the set of letters in the entire alphabet. The reader may well wonder if there is an ultimate universal set which contains everything. The short answer is ‘no’ and we refer you once again to Russell’s Paradox. The usual triptych of Venn Diagrams indicating generic sets \(A\) and \(B\) along with \(A \cap B\) and \(A \cup B\) is given below.

The one major limitation of Venn Diagrams is that they become unwieldy if more than four sets need to be drawn simultaneously within the same universal set. This idea is explored in the Exercises.

1 Sets of Real Numbers

The playground for most of this text is the set of Real Numbers. Much of the “real world” can be quantified using real numbers: the temperature at a given time, the revenue generated by selling a certain number of products and the maximum population of Sasquatch which can inhabit a particular region are just three basic examples. A succinct, but nonetheless incomplete ^[6] (Math pun intended!) definition of a real number is given below.

Certain subsets of the real numbers are worthy of note and are listed below. In fact, in more advanced texts, ^[7] (See, for instance, Landau’s Foundations of Analysis.) the real numbers are constructed from some of these subsets.

1. The Natural Numbers: \(\mathbb {N}= \{ 1, 2, 3, \ldots \}\) The periods of ellipsis ‘\(\ldots \)’ here indicate that the natural numbers contain \(1\), \(2\), \(3\) ‘and so forth’.
2. The Whole Numbers: \(\mathbb {W} = \{ 0, 1, 2, \ldots \}.\)
3. The Integers: \(\mathbb Z=\{ \ldots , -3, -2, -1, 0, 1, 2, 3, \ldots \} = \{ 0, \pm 1, \pm 2, \pm 3, \ldots \}.\) ^[8] (The symbol \(\pm \) is read ‘plus or minus’ and it is a shorthand notation which appears throughout the text. Just remember that \(x = \pm 3\) means \(x = 3\) or \(x = -3\).)

4. The Rational Numbers: \(\mathbb {Q}=\left \{\frac {a}{b} \, | \, a \in \mathbb Z \, \mbox {and} \, b \in \mathbb Z \, \mbox {where} \, b \neq 0\right \}\). Rational numbers are the ratios of integers where the denominator is not zero. It turns out that another way to describe the rational numbers ^[9] (See Section ??.) is:

   \[\mathbb {Q}=\{x\,|\,\mbox {$x$ possesses a repeating or terminating decimal representation}\}\]

5. The Irrational Numbers: \(\mathbb P = \{x\,|\,\mbox {$x \in \mathbb {R}$ but $x \notin \mathbb {Q}$}\}\). ^[10] (Examples here include number \(\pi \) (See Section ??), \(\sqrt {2}\) and \(0.101001000100001\ldots \).) That is, an irrational number is a real number which isn’t rational. Said differently,

   \[\mathbb P = \{x\,|\,\mbox {$x$ possesses a decimal representation which neither repeats nor terminates}\}\]

Note that every natural number is a whole number which, in turn, is an integer. Each integer is a rational number (take \(b =1\) in the above definition for \(\mathbb {Q}\)) and since every rational number is a real number ^[11] (Thanks to long division!) the sets \(\mathbb {N}\), \(\mathbb {W}\), \(\mathbb Z\), \(\mathbb {Q}\), and \(\mathbb {R}\) are nested like Matryoshka dolls. More formally, these sets form a subset chain: \(\mathbb {N} \subseteq \mathbb {W} \subseteq \mathbb Z \subseteq \mathbb {Q}\subseteq \mathbb {R}\). The reader is encouraged to sketch a Venn Diagram depicting \(\mathbb {R}\) and all of the subsets mentioned above.

It is time to put all of this together in an example.

As you may recall, we often visualize the set of real numbers \(\mathbb {R}\) as a line where each point on the line corresponds to one and only one real number. Given two different real numbers \(a\) and \(b\), we write \(a < b\) if \(a\) is located to the left of \(b\) on the number line, as shown below.

While this notion seems innocuous, it is worth pointing out that this convention is rooted in two deep properties of real numbers. The first property is that \(\mathbb {R}\) is complete. This means that there are no ‘holes’ or ‘gaps’ in the real number line. ^[21] (Alas, this intuitive feel for what it means to be ‘complete’ is as good as it gets at this level. Completeness is given a much more precise meaning later in courses like Analysis and Topology.) Another way to think about this is that if you choose any two distinct (different) real numbers, and look between them, you’ll find a solid line segment (or interval) consisting of infinitely many real numbers. The next result tells us what types of numbers we can expect to find.

Between any two distinct real numbers, there is at least one rational number and one irrational number. It then follows that between any two distinct real numbers there will be infinitely many rational and infinitely many irrational numbers.

The root word ‘dense’ here communicates the idea that rationals and irrationals are ‘thoroughly mixed’ into \(\mathbb {R}\). The reader is encouraged to think about how one would find both a rational and an irrational number between, say, \(0.9999\) and \(1\). Once you’ve done that, try doing the same thing for the numbers \(0.\overline {9}\) and \(1\). (‘Try’ is the operative word, here. ^[22] (Again, see Section ?? for details.) )

The second property \(\mathbb {R}\) possesses that lets us view it as a line is that the set is totally ordered. This means that given any two real numbers \(a\) and \(b\), either \(a < b\), \(a > b\) or \(a = b\) which allows us to arrange the numbers from least (left) to greatest (right). This property is given below.

If \(a\) and \(b\) are real numbers then exactly one of the following statements is true:

Segments of the real number line are called intervals. They play a huge role not only in this text but also in the Calculus curriculum so we need a concise way to describe them. We start by examining a few examples of the interval notation associated with some specific sets of numbers.

Set of Real Numbers	Interval Notation	Region on the Real Number Line

As you can glean from the table, for intervals with finite endpoints we start by writing ‘left endpoint, right endpoint’. We use square brackets, ‘\([\)’ or ‘\(]\)’, if the endpoint is included in the interval. This corresponds to a ‘filled-in’ or ‘closed’ dot on the number line to indicate that the number is included in the set. Otherwise, we use parentheses, ‘\((\)’ or ‘\()\)’ that correspond to an ‘open’ circle which indicates that the endpoint is not part of the set. If the interval does not have finite endpoints, we use the symbol \(-\infty \) to indicate that the interval extends indefinitely to the left and the symbol \(\infty \) to indicate that the interval extends indefinitely to the right. Since infinity is a concept, and not a number, we always use parentheses when using these symbols in interval notation, and use the appropriate arrow to indicate that the interval extends indefinitely in one or both directions. We summarize all of the possible cases in one convenient table below. ^[23] (The importance of understanding interval notation in this book and also in Calculus cannot be overstated so please do yourself a favor and memorize this chart.)

    Let \(a\) and \(b\) be real numbers with \(a<b\).

Set of Real Numbers	Interval Notation	Region on the Real Number Line

Intervals of the forms \((a, b), (-\infty , b)\) and \((a, \infty )\) are said to be open intervals. Those of the forms \([a, b], (-\infty , b]\) and \([a, \infty )\) are said to be closed intervals.

Unfortunately, the words ‘open’ and ‘closed’ are not antonyms here because the empty set \(\emptyset \) and the set \((-\infty , \infty )\) are simultaneously open and closed ^[24] (You don’t need to worry about that fact until you take an advanced course in Topology.) while the intervals \((a, b]\) and \([a, b)\) are neither open nor closed. The inclusion or exclusion of an endpoint might seem like a terribly small thing to fuss about but these sorts of technicalities in the language become important in Calculus so we feel the need to put this material in the Precalculus book.

We close this section with an example that ties together some of the concepts presented earlier. Specifically, we demonstrate how to use interval notation along with the concepts of union and intersection to describe a variety of sets on the real number line. In many sections of the text to come you will need to be fluent with this notation so take the time to study it deeply now.