1.1 Sets & Numbers

Introduction

The Smithsonian in Washington, DC is one of the largest museums in the world, housing vast collections of rare and historically significant objects.

The Air and Space collection, for example, holds the 1903 Wright Flyer, the first successful airplane, and the Apollo 11 Command Module from the first mission to the moon.

Notable items in other collections include Abraham Lincoln’s top hat, an original Kermit the Frog puppet, and the famous Hope Diamond.

With over 200 million fossils, specimens, and treasures, these collections play a crucial role in preserving our nation's history and culture, serving as invaluable resources for researchers and scientists.

Collections are equally vital in mathematics, although you seldom find them in museum exhibits.

Sets

In mathematics, collections are usually called sets. Sets are the foundational building blocks of modern mathematics.

To create a set, you simply need a way to decide if an object belongs in the set or not.

One way to do that is to list everything in the set. For example, traffic lights only have three colors: red, yellow, and green. We can list these colors to form a set: $\lbrace \text{red, yellow, green} \rbrace$. Items in a set are always written inside curly braces $\lbrace \thinspace \thinspace \rbrace$.

Another way to represent a set is by drawing a big circle and writing each item of the set inside the circle.

Visual diagrams like this will be helpful in later sections where we investigate relationships between sets.

Set-Builder Notation

Sometimes it's hard to list everything in a set. For instance, whiskers on kittens, bright copper kettles and warm woolen mittens might be a few of my favorite things, but listing all my favorites would be impossible.

Fortunately, instead of listing everything in a set, we can simply describe the properties that the items share. This is called set-builder notation.

Using set-builder notation, the set of "all of my favorite things" could be written as

[ \left\lbrace x \thinspace \vert \thinspace x \text{ is one of my favorite things} \right\rbrace ]

The vertical bar symbol $\vert$ means "such that" or "where". If we put the set above into words we would read it as "the set of all $x$ where $x$ is one of my favorite things."

Combining Sets

Sets can be combined in two different ways. The first way is to group everything from both sets together into one big set. This is called the union of the two sets. In the figure below you can form the union of the two sets by moving the blue slider.

There is a special symbol we use for unions: $\bigcup$. If we have two sets, $A$ and $B$, then we write their union as $A \bigcup B$

[ A \bigcup B = \left\lbrace x \thinspace \vert \thinspace x \text{ is in } A \text{ or } x \text{ is in }B \right\rbrace ]

The second way to combine sets is to only consider items that the two sets have in common. This is called the intersection of the sets.

The intersection of two sets also has its own symbol: $\bigcap$.

[ A \bigcap B = \left\lbrace x \thinspace \vert \thinspace x \text{ is in } A \text{ and } x \text{ is in }B\ \right\rbrace ]

If the intersection has nothing in it we say it is empty and use the symbol $\varnothing$ to represent the empty set.

Counting

It might be hard to imagine, but people have not always had numbers. The numbers we use today are actually the result of thousands of years of innovation and invention.

Surprisingly, sets actually came before numbers. To see why this is the case, take a look at the following sets. What do they have in common?

These sets are similar because they are the same size. Even though the contents are very different, the quantity of items in each set is identical. In other words, they have the same number of items.

It's this need to count the objects in a set that sparked the development of numbers.

Development of Numbers

There's little doubt that fingers were among the earliest counting tools. In fact, in English the words eleven and twelve come from the Old English words for "one left over" and "two left over", indicating how many would be left over after counting to ten with your fingers.

Over time people invented other ways to keep track of their flocks, herds, and commodities. They cut tally marks on sticks, made piles of pebbles, and even carved figurines to look like sheep, jars of oil, baskets of grain, and so on. Tally sticks and figurines from the Swiss Alpine Museum. Photo by Sandstein.

In Mesopotamia, about 10,000 years ago, Sumerians made clay tokens shaped to resemble the items they were exchanging, one token for each item. Soon these tokens were replaced by shapes drawn on clay tablets.

Eventually special symbols, or numerals, were created to represent specific quantities. It was at that point, some 5000 years ago, that the first number system was born. This 5000 year-old clay tablet is one of the first artifacts in history to display numerical symbols. It records how much beer should be paid to a particular worker for his labor. Photo courtesy of the British Museum

The Sumerian system did not look like the one we use today. They counted using groups of 60 (which is why we have 60 minutes in an hour) and used different symbols than what we are used to.

The number system we now use worldwide originated in India and was later adopted by Arab scholars, which is why it is called the Hindu-Arabic system.

It wasn't until the 15th and 16th centuries that this system caught on in Europe. Since then, Hindu-Arabic numerals have become the standard numerical representation used globally for counting, record-keeping, and mathematical calculations, havng spread through international trade, scholarship, and cultural exchange.

Natural Numbers and Integers

The numbers we are most familiar with are the counting numbers $1, 2, 3, 4, ...$. The set of counting numbers is also known as the set of natural numbers and is represented by the symbol $\mathbb{N}$. In set notation we would say

[ \mathbb{N} = \left\lbrace 1,2,3,...\right\rbrace ]

You might already know that adding two natural numbers always results in another natural number. For example, $3$ and $5$ are both natural numbers and so is $3+5=8$.

But what if we subtract one natural number from another, is the result always in $\mathbb{N}$? Clearly not, since subtraction can create negative values.

The set that combines $\mathbb{N}$ with $0$ and all the negative natural numbers $ \left\lbrace ..., -3, -2, \right\rbrace$ is called the set of integers and is represented by the symbol $\mathbb{Z}$.

[ \mathbb{Z} = \left\lbrace ...,-3,-2,-1,0,1,2,3,... \right\rbrace ]

Notice that we can add and subtract integers as much as we like and the result is always an integer.

Rational Numbers

While multiplying two integers always results in another integer, division is more complicated.

If $p$ and $q$ are integers, then the fraction $\frac{p}{q}$ could be an integer, but it doesn't have to be. For instance $\frac{6}{2}=3$ is an integer, but $\frac{5}{2}$ is not.

The set that contains all the fractions, or ratios, of integers, is called the set of rational numbers, or $\mathbb{Q}$.

[ \mathbb{Q} = \left\lbrace \frac{p}{q} \thinspace \bigg| \thinspace {p} \text{ and } {q} \text{ are integers with } q\neq0 \right\rbrace. ]

Since division by $0$ is not well defined, fractions with a denominator of $0$, such as $\frac{4}{0}$ and $\frac{-2}{0}$, are not included in the set of rational numbers.

It is important to note that since any integer can be written as a fraction (for example $5 = \frac{5}{1}$), the set of rational numbers includes all of the integers as well.

Fractions can be converted into decimals by dividing the top number (the numerator) by the bottom number (the denominator). When you do this one of two patterns will occur. Either the divsion will end after a certain number of decimal places, such as $\frac{10}{5}=2$ and $\frac{3}{16}=0.1875$, or it will repeat forever like $\frac{1}{3}=0.\overline{3}$ and $\frac{1}{7}=0.\overline{142857}$.

The set of rational numbers is special because you can add, subtract, multiply and divide as much as you want and never end up with a different type of number---the answer is always a rational number.

Real Numbers

Since the rational numbers are closed under addition, subtraction, multiplication and division, it might come as a surprise that other types of numbers exist and are essential to mathematics. The discovery of one new type of number is attributed to an associate of Pythagoras whose name was Hippasus of Metapontum.

According to legend, Hippasus was trying to measure the diagonal of a 1 by 1 square . He knew from the Pythagorean theorem that the answer was $\sqrt{2}$, but had been unable to find a rational number that was equal to $2$ when you square it.

Eventually he proved that $\sqrt{2}$ cannot be written as a ratio of any two integers and concluded that it must belong to a new set of irrational numbers.

Some accounts say that the Pythagoreans were so shocked by Hippasus' discovery that they drowned him in the Mediterranean Sea to try and hide irrational numbers from the world.

That attempt failed, and we now know that there are actually far more irrational numbers than rational numbers. Some irrational numbers are famous and have their own special symbols, like $pi$, but most are written as expressions involving operations like $\sqrt{2}$.

Irrational numbers cannot be expressed exactly as fractions or finite decimals since their decimal expansions never end or repeat. In many cases we use a calculator to approximate irrational numbers by rounding to a specified number decimal places. For example, $\sqrt{2}\approx 1.4142$ rounded to $4$ decimal places.

The union of the sets of rational and irrational numbers is called the set of real numbers, $\mathbb{R}$. In other words, the set of real numbers includes every type of number we've discussed so far.

For example, $-0.5$, $\sqrt{2}$, $\frac{7}{3}$, $-0.248$, and $8.\overline{261}$ are all real numbers.

The set of real numbers is closed under all standard operations including powers and some roots. In fact, you can do just about anything to a real number and the result will still be a real number, as long as you do not take the square root of a negative value, something we will look into in a later chapter, or divide by $0$.

The Real Number Line

It is often helpful to visualize the set of real numbers with a number line, which is just a straight line with marks representing the location of numbers.

Positive numbers are always on the right side of zero while negative numbers are always on the left.

Every real number has a place on the number line, and every location on a number line is assumed to correspond to a real number. Here are a few real numbers and their relative locations on the number line.

Intervals of Real Numbers

The set of all values in between two numbers is called an interval. For instance, all of the numbers between $0$ and $3$ make an interval.

There are several ways to describe this interval. One way is to use inequalities and write it as a set.

[\lbrace x \thinspace \vert \thinspace 0<x<3 \rbrace]

Another option is to color the interval on a numberline

Open circles indicate that the numbers $0$ and $3$ are not included in the interval.

A third option is to use interval notation, which is a written abbreviation of the number line. In this example the interval notation would be

[(0 , 3)]

Notice that this mirrors the number line by having the lowest number on the left, with parenthesis taking the place of the open circles.

Other intervals might include an end point or even extend out toward infinity. For example, the interval of all numbers greater than or equal to $-2$ can be described in any of the following ways

• set notation: [\lbrace x\vert-2\leq x<\infty \rbrace]
• number line:
• interval notation: [[ -2\thinspace, \thinspace \infty)]

There are a few subtleties here that deserved to be highlighted.

First, notice that there are three ways to indicate that an endpoint is included. In set notation, the inequality used must have an equal sign ($\leq$ or $\geq$). On a number line, the circle marking the point is filled in. And in the interval notation, the parenthesis for that point changes to a square bracket.

Second, to indicate that an interval continues out toward infinity we use an arrow pointing in the appropriate direction. Since nothing equals infinity we must always use a parenthesis when writing that part of the interval.

Combining Intervals

Since intervals are sets of real numbers, they can be combined using unions and intersections.

The union is result of combining the two intervals together, while the intersection is only the part where the intervals overlap each other. A few examples will illustrate the idea.

Suppose we have the following intervals. What will the union look like? Move the blue slider to see the union.

Once we've identified the union on the number line, it's much easier to write it in interval notation.

[(-1, \infty) \bigcup [-3,0 ) = [-3, \infty)]

The intersection of the same two intervals is smaller, since it is only where they overlap. You can explore the intersection in the figure below.

Now that we can see the intersection visually, the notation follows naturally.

[(-1, \infty) \bigcap [-3,0 ) = (-1, 0)]

Order of operations

In elementary school students are often asked to add several numbers together, such as $4 + 10 + 5 = 19$. What is seldom discussed is the fact that we can't technically add three numbers at once.

Addition, like the other basic operations, is a "binary" operation--it only works on two numbers a time. To avoid confusion, when multiple operations are needed there is a standard order that should be followed.

The order of operations we use today is sometimes remembered by the acronym P E MD AS, which stands for

1. Parentheses (including brackets and fraction bars)
2. Exponents (including roots)
3. Multiplication and Division
4. Addition and Subtraction

When several of these operations are combined, operations higher on the list should be dealt with before those lower on the list.

And if an expression has more than one operation with equal priority (like multiplication and division or addition and subtraction) then you work with them in the order they appear from left to right. For instance, if we have $20 \div 5 \times 2$, then we start from the left and first do $20 \div 5$, which is $4$, and then multiply that by $2$, giving us $8$.

Additional Practice

Want more practice? Try these optional exercises powered by MyOpenMath. These exercises are purely for practice and are not part of your graded homework - you can attempt them as many times as you'd like!