4.2 Power Sets and Cartesian Products

Objectives:

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Discover how we can build larger sets from other sets using Cartesian products and power sets

4.2.1 Cartesian Products

Example 4.2.1.

We often see the set $\mathbb{R}^{2}$ as the set of all $(x,y)$ -coordinates, also known as the $x y$ -plane. In what sense is $\mathbb{R}^{2}$ the “square” of the real numbers? Where does the number “2” come into play when defining elements of $\mathbb{R}^{2}$ ?

Is it possible to define $[0,1]^{2}$ ? If so, how would you define it? Draw a picture of this region.

Cartesian products are especially useful when dealing with functions where several variables are involved. Before this class is over we will see several other uses for them as well… we will define relations and even functions themselves as subsets of Cartesian products!

Definition 4.2.2.

Let $A$ and $B$ be sets. The Cartesian product $\times$ (LaTeX $\backslash$ times) of $A$ and $B$ is the set

A\times B:=\{(x,y)\mid x\in A\text{ and }y\in B\},

where $(x,y)$ denotes the ordered pair containing $x$ and $y$ in that order.

Example 4.2.3.

List all elements in the set $\{1,2\}\times\{1,\pi,\{0\}\}$ .

It’s possible to repeat this process of taking Cartesian products over and over to get larger and larger sets. For example, if we are considering a rectangular prism of length 3, width 2, and height 1, we might describe this as the following subset in $\mathbb{R}^{3}$ : $[0,3]\times[0,2]\times[0,1]$ .

To be specific about this definition and extend it to work for any finite number of Cartesian products, we define:

Definition 4.2.4.

For each $n\in\mathbb{N}$ , we define an $n$ -tuple to be an ordered list of $n$ elements of the form $(a_{1},a_{2},\dots,a_{n})$ . We refer to $a_{i}$ as the $i$ th component (or coordinate) of $(a_{1},a_{2},\dots,a_{n})$ .

Given $n$ sets $A_{1},A_{2},\dots,A_{n}$ , we define the Cartesian product of these $n$ sets to be the set of all $n$ -tuples where the $i$ th coordinates $a_{i}$ come from the set $A_{i}$ . That is,

\prod_{i=1}^{n}A_{i}:=A_{1}\times\cdots\times A_{n}:=\{(a_{1},\dots,a_{n})\mid a% _{j}\in A_{j}\text{ for all }1\leq j\leq n\}.

(The Cartesian Product is written as $\backslash$ prod in LaTeX.) The special case of the set

\underbrace{A\times\cdots\times A}_{n\text{ factors}}

is referred to as $A^{n}$ .

You will see us start to do fewer and fewer proofs using new concepts in class; indeed, you will have to do more of the proofs utilizing these concepts on your own now. This is important to strengthen your ability to verify mathematical statements yourself. It is a scary step to take… it scared me too when I was at this point in the class. Here is a quote from Leon Henkin, an American logician at UC Berkeley, that addresses why we are making this step:

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.

Example 4.2.5.

Interpret this quote in your own words.

Theorem 4.2.6.

Let $A, B, C,$ and $D$ be sets. If $A\subseteq C$ and $B\subseteq D$ , then $A\times B\subseteq C\times D$ .

Example 4.2.7.

Let $A$ and $B$ be sets, each within some universe of discourse. Conjecture a way to rewrite $(A\times B)^{C}$ that meaningfully involves $A^{C}$ and $B^{C}$ . For homework, prove your conjecture. (Hint: what is $([0,1] \times [0,1])^{c}$ ? Draw a picture.)

(More space for Example 4.2.7.)

4.2.2 Power Sets

Example 4.2.8.

The letters of the word “MAROON” are put into a hat, and a student chooses a letter at random. Write out the set of all letters they could choose this way.

What is the set of all vowels the student could choose?

What about the set of all letters in the second half of the alphabet that the student could choose?

This time, the student can choose any number of letters in the hat, including all of them or none of them. How many possible combinations of letters can the student have? How do you know?

What if there are $n$ distinct letters (not neceesarily "M", "A", "R", "O", "N") in the hat that the student could choose from? How many possible combinations of letters can the student have?

In the world of probability, it is necessary to calculate how many possible subsets satisfy a certain outcome. If the example from the last page was a game where choosing a vowel could win or lose you some money, you might be quite interested to know how likely the outcome would be of winning!

All of these subsets live in the power set of a set $S$ . For the definition below, think of the set $S$ as the set $\{M,A,R,O,N\}$ from the previous example; the set $\mathcal{P}(S)$ is the set of all subsets of this set, which we found in the second-to-last example had 32 elements.

Definition 4.2.9.

If $S$ is a set, then the power set of $S$ is the set of subsets of $S$ . The power set is denoted $\mathcal{P}(S)$ (LaTeX $\backslash$ mathcal $\{$ P $\}$ ).

Theorem 4.2.10.

Let $S$ be a set with $n$ elements. Using induction, prove that $\mathcal{P}(S)$ has $2^{n}$ elements.

(More space for Theorem 4.2.10.)