4 Sets

4.1 The Language of Sets

Objectives:

• Discover the fundamental building blocks of mathematics

• Create examples involving important sets in mathematical proof

• Learn DeMorgan’s Laws and distributive properties of sets

From this point onward we will begin setting this course up as a class in formal mathematics. In your classes before we have been focused nearly entirely on examples. From here on we will focus much more on the definitions, theorems, and proofs. This does not mean that examples have no place in mathematics - we will still use them to get an idea of what our definitions and theorems are all about! But these will take a backseat to the more overarching concepts in each section.

Here are some tips about how to navigate a rigorous class in mathematics:

• Remember your fundamentals. Everything you have learned up to this point will assist you. You will use several different kinds of proving techniques in each day of lecture.

• When in doubt, come up with an example. If you are uncertain about what a definition says or what a theorem means, come up with a few different examples and see if you can make sense of what the definition/theorem is saying. As you continue to strengthen in mathematics, try to “break” the theorems you see and come up with examples that might make it untrue. (This will be difficult now, so just use common examples for now.)

• Read along with others in the class. You will see that the textbook and the notes supply two different perspectives on the same concepts, but often two perspectives is not enough to get a full understanding. You need to hear from your peers and get their perspective. Likewise, your peers need to hear from you. Discuss mathematics with one another, and this will certainly make you stronger.

We begin by discussing sets, which are the foundational tool that all of modern-day mathematics is built upon. Numbers, functions, rectangles, polygons, vectors, and more can all be written in terms of sets and their elements.

Definition 4.1.1.

A set is a collection of objects called elements. If $A$ is a set and $x$ is an element of $A$, we write $x\in A$ (LaTeX $\backslash$in). Otherwise, we write $x\notin A$ (LaTeX $\backslash$notin).

The set containing no elements is called the empty set and is denoted by either $\mathrm{\varnothing }$ (LaTeX $\backslash$varnothing) or $\{\}$. Any set that contains at least one element is referred to as a nonempty set.

The most common way to express sets is in terms of set-builder notation:

$$S=\{x\in A\mid P(x)\},$$

where $P(x)$ is some predicate statement involving $x$.

The first part $x\in A$ denote what type of $x$ is being considered. For example, the set $\{x\in \mathbb{N}|x\text{ is even and }x\ge 8\}$ requires that the number $x$ be a natural number before continuing to the set’s conditions.

The second part puts conditions on the $x$ being considered. In the set $\{x\in \mathbb{N}|x\text{ is even and }x\ge 8\}$, the conditions are two-fold: $x$ must be even, and $x$ must be a number greater than or equal to 8. This is a more formal notation for the set below, using what is known as roster notation:

$$\{8,10,12,14,\mathrm{\dots }\}$$

This notation supposes a pattern that is not that is not clarified in the set’s definition. It is easier to prove statements about a set when that set has already been written in set-builder notation (such as the rational numbers below).

The following sets are very common in mathematics:

• $\mathbb{N}:=\{1,2,3,\mathrm{\dots }\}$ is the set of natural numbers. Some mathematicians consider 0 to be a natural number; we will not.

• $\mathbb{Z}:=\{\mathrm{\dots },-3,-2,-1,0,1,2,3,\mathrm{\dots }\}$ is the set of integers.

• $\mathbb{Q}:=\{a/b\mid a\in \mathbb{Z},b\in \mathbb{N}\}$ is the set of rational numbers. (Note that my writing of $\mathbb{Q}$ differs from the book’s. However, the two sets are identical. There are multiple ways to write the same set. In a while we will prove these two sets are the same.)

• $ℝ$ denotes the set of real numbers.

Example 4.1.2.

1. Use roster notation to provide a description of the elements of the set below.

   $$\left\{m\in ℝ\right|m=1-\frac{1}{n}\text{ where }n\in \mathbb{N}\}.$$

   Blank space for you to write your work.

2. Write the following set description in terms of set-builder notation: “the set of all even integers”. (There is more than one correct answer… can you find multiple answers?)

   Blank space for you to write your work.

We can make sets out of much more than numbers - we can take sets of vectors, functions, variables, and any combination of these three! Sometimes we will refer to sets using arbitrary variables $a,b,c,\mathrm{\dots }$ to denote that the elements can be of any variety.

When we are discussing sets containing numbers, we will be specific: we will say they are subsets of any of the sets we have discussed above.

Definition 4.1.3.

If $A$ and $B$ are sets, then we say that $A$ is a subset of $B$, written $A\subseteq B$ (LaTeX $\backslash$subseteq) provided that every element of $A$ is an element of $B$.

For the purposes of learning about what sets can do, we will stick to numbers for now. Notice that each of the intervals defined below are subsets of the real numbers (how can you tell?):

Definition 4.1.4.

For $a,b\in ℝ$ with , we define the following sets, referred to as intervals.

2. $[a,b]:=\{x\in ℝ\mid a\le x\le b\}$

6. $(-\mathrm{\infty },\mathrm{\infty }):=ℝ$

Intervals of the form $(a,b),(-\mathrm{\infty },b),(a,\mathrm{\infty }),$ and $(-\mathrm{\infty },\mathrm{\infty })$ are called open intervals while $[a,b]$ is referred to as a closed interval.

A bounded interval is any interval of the form $(a,b)$, $[a,b)$, $(a,b]$, and $[a,b]$.

For bounded intervals, $a$ and $b$ are called the endpoints of the interval.

Here and elsewhere, whenever we write intervals with both $a$ and $b$ as above, we will assume .

Example 4.1.5.

Give three meaningfully different examples of infinite sets that are not intervals.

Blank space for you to write your work.

4.1.1 Proving with Sets

We are now ready to begin proving with sets. For the rest of this subsection, let’s make the following definitions for subsets $A$, $B$, and $C$ of the real numbers:

$$A=[-3,-1),B=(-2.5,2),C=(-2,0].$$

Example 4.1.6.

Is $C$ a subset of $B$? Explain your reasoning using a number line.

Blank space for you to write your work.

In future problems we will be unable to use a number line to neatly show one set is a subset of another. We need a way to write a formal proof for this statement.

A Skeleton Version of a Direct Proof for $C\subseteq B$.

Guidelines	Example
Let $x$ be an element of $C$.	Assume that $x$ is in $C$, or that $x$ an element of $(-2,0]$.
Use definitions to describe relevant properties of $x$. (There is sometimes more than one way to do this.)	Then .
(Recommended) State what you need to show in order to conclude $x\in B$.	We want to show that $x\in B$, or that .
Connect the two pieces: what you have ($x\in C$) and what you need ($x\in B$).	???
Conclude $B$.	Since $x$ was an arbitrary element of $C$ and was found to also be in $B$, we may conclude that $C\subseteq B$, as desired.

Example 4.1.7.

Fill in the blank box in the skeleton version for $C\subseteq B$ to complete the proof of the statement from Example 4.1.6.

Blank space for you to write your work.

You may notice that this models proving a conditional proposition using direct methods. This leads to the question: is there a version of the contrapositive for direct proofs for sets? We can absolutely say yes!

To do this, we have to understand what it means to not be in a set. Say we are not in the set $A$ from the previous page. What are examples of elements not in $A$? Certainly positive numbers are not in $A$. But what about weird stuff like vectors and functions? Those aren’t in $A$ either, right?

We must rein in our universe of discourse to be specifically about real numbers when considering elements not in $A$. This leads us to our next definition:

Definition 4.1.8.

Let $A$ be a set in some universe of discourse $U$. Then the complement of $A$ (relative to $U$) is the set

$$A^c:=U\setminus A=\{x\in U\mid x\notin A\}.$$

In LaTeX, $\setminus$ is written as $\backslash$setminus.

Theorem 4.1.9.

Prove that, if $A\subseteq B$, then $B^c\subseteq A^c$.

Blank space for you to write your work.

There is also a version of biconditional proof: if we show that $A\subseteq B$ and $B\subseteq A$, then both sets are equal!

It is just as important that we learn how to show that something is not a subset as we show something is a subset. To do this, we observe how the definition of “is a subset of” can be written using quantifiers.

We say $A\subseteq B$ if every element of $A$ is an element of $B$.

$$A\subseteq B\iff (\forall x\in A)(x\in B).$$

This means that to say that something is not a subset of $A$ entails negating this statement:

$$A⊈B\iff (\exists x\in A)(x\notin B).$$

Specifically, to show that something is not a subset is a “there exists” statement, not a “for all” statement!

Example 4.1.10.

Let $A=\{n\in \mathbb{Z}\mid (\exists k\in \mathbb{Z})(n=4k+1)\}$ and $B=\{n\in \mathbb{Z}\mid n\text{ is odd}\}$.

Show that $A\subseteq B$ but $B⊈A$.

Blank space for you to write your work.

Example 4.1.11.

For each of the set operations given below, describe the set using the Venn diagram provided.

Definition 4.1.12.

Let $A$ and $B$ be sets in some universe of discourse $U$.

1. The union $\cup$ (LaTeX $\backslash$cup) of the sets $A$ and $B$ is

   $$A\cup B:=\{x\in U\mid x\in A\text{ or }x\in B\}.$$

2. The intersection $\cap$ (LaTeX $\backslash$cap) of the sets $A$ and $B$ is

   $$A\cap B:=\{x\in U\mid x\in A\text{ and }x\in B\}.$$

3. The set difference $A\setminus B$ of the sets $A$ and $B$ is

   $$A\setminus B:=\{x\in U\mid x\in A\text{ and }x\notin B\}.$$

(a) (b) (c)

Example 4.1.13.

Fill in the Venn diagrams below to represent ${(A\cup B)}^c$ and $A^c\cap B^c$.

${(A\cup B)}^c$

$(A^c\cap B^c)$

What do you notice?

Blank space for you to write your work.

To prove two sets $A$ and $B$ are equal, we must show that each set is a subset of the other. We will prove this using the following two-step process:

1. First show $A\subseteq B$.

2. Then show $B\subseteq A$.

Theorem 4.1.14.

Using a know-show table, prove one of DeMorgan’s Laws: for sets $A$ and $B$, ${(A\cup B)}^c=A^c\cap B^c$.

Blank space for you to write your work.

(More work for Theorem 4.1.14.)

Example 4.1.15.

Prove the following statement:

$$\left\{\frac{a}{b}\right|a,b\in \mathbb{Z},b\ne 0\}=\left\{\frac{a}{b}\right|a\in \mathbb{Z},b\in \mathbb{N}\}.$$

Either set can be used in the definition of the rational numbers $\mathbb{Q}$.

Blank space for you to write your work.