# MAT 112 Integers and Modern Applications for the Uninitiated

## Section6.2Cartesian Products

A Cartesian product of two sets is a new set that is constructed from the two sets. In order to define Cartesian products, we need to define a mathematical object called an ordered pair.

### Definition6.11.

An ordered pair is an ordered list of two mathematical objects, $$a$$ and $$b\text{,}$$ written as $$(a,b)\text{.}$$ The objects in an ordered pair are called components. The object $$a$$ is the first component of $$(a,b)\text{,}$$ and the object $$b$$ is the second component of $$(a,b)\text{.}$$

### Definition6.12.

Let $$A$$ and $$B$$ be sets. The Cartesian product of $$A$$ and $$B\text{,}$$ denoted $$A \times B\text{,}$$ is the set of ordered pairs $$(a,b)\text{,}$$ where $$a \in A$$ and $$b \in B\text{.}$$
The Cartesian product of two sets $$A$$ and $$B\text{,}$$ formulated in set-builder notation, is
\begin{equation*} A \times B := \set{(a,b) \mid a \in A \text{ and } b \in B}\text{.} \end{equation*}
Complete the definition of Cartesian products in Checkpoint 6.13.

### Checkpoint6.13.Definition of Cartesian product.

Complete the following:
Let $$A$$ and $$B$$ be sets.
The
• select
• intersection
• union
• difference
• sum
• Cartesian product
• complement
of the sets $$A$$ and $$B\text{,}$$ denoted by $$A\times B\text{,}$$ is the set of all
• select
• unordered pairs
• ordered pairs
• sets
• numbers
$$(a,b)$$ where $$a$$ is
• select
• related to
• an element of
• a subsets of
• not related to
• not an element of
• a proper subset of
• equal to
• not equal to
the set $$A$$
• select
• and
• or
$$b$$ is
• select
• related to
• an element of
• a subsets of
• not related to
• not an element of
• a proper subset of
• equal to
• not equal to
the set $$B\text{.}$$
To form the Cartesian product $$A \times B\text{,}$$ we pair each element of $$A\text{,}$$ placed in the first component of the ordered pair, with each element of $$B\text{,}$$ placed in the second component of the ordered pair.

### Example6.14.Cartesian products in roster form.

Let $$A = \set{0,1}\text{,}$$ and let $$B = \set{4,5,6}\text{.}$$ Then,
\begin{equation*} A \times B = \set{(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6)}\text{,} \end{equation*}
and
\begin{equation*} B \times A = \set{(4, 0), (4, 1), (5, 0), (5, 1), (6, 0), (6,1)}\text{.} \end{equation*}

### Problem6.15.

Let $$A=\{1,2,3\}$$ and let $$B=\{-50\}\text{.}$$ Give the set $$A\times B$$ in roster form.
Solution.
The set $$A\times B$$ contains all ordered pairs whose first entry is an element of the set $$A$$ and whose second entry is an element of the set $$B\text{.}$$ We write ordered pairs whose first entry is $$c$$ and whose second entry is $$d$$ as $$(c,d)\text{.}$$ We get
\begin{equation*} A\times B= \{(1,-50),(2,-50),(3,-50)\} \end{equation*}
In the next problem a Cartesian product is given in set builder notation.

### Problem6.16.

Let $$A = \{12,13,34\}\text{.}$$ Give $$\{(a,a \fmod 5)\mid a \in A\}$$ in roster form.
Solution.
We find all pairs whose first entry is an element $$a$$ of the set $$A$$ and whose second entry is $$a \fmod 5\text{.}$$ We get
\begin{align*} \{(a,a \fmod 5) \mid a \in A\} \amp = \{(12,12 \fmod 5), (13,13 \fmod 5), (34,34\fmod 5)\}\\ \amp =\{(12,2), (13,3), (34,4)\}\text{.} \end{align*}
In Checkpoint 6.17 determine all the elements of a Cartesian product that is given in set-builder notation.

### Checkpoint6.17.Cartesian product from set-builder to roster.

Let $$A = \lbrace 5, 6, 7, 8 \rbrace\text{.}$$ Give the set in roster form.
$$\lbrace ( a, 3\cdot a ) \mid a \in A \rbrace =$$$$\lbrace$$$$\rbrace$$
[although it would be mathematically correct to list elements multiple times, this problem is marked wrong if you do so.]
To determine when two Cartesian products are equal we need to know when two ordered pairs are equal.

### Definition6.18.

Let $$A$$ and $$B$$ be sets. Saying that $$(a,b)\in A\times B$$ and $$(c,d)\in A\times B$$ are equal means that $$a=c$$ and $$b=d\text{.}$$
When $$(a,b)$$ and $$(c,d)$$ are equal, we write $$(a,b)=(c,d)\text{.}$$
When $$(a,b)$$ and $$(c,d)$$ are not equal, we write $$(a,b)\ne(c,d)\text{.}$$
So, two ordered pairs are equal if they have matching first components and matching second components. The fact that the elements $$(a,b)$$ of $$A \times B$$ are called ordered pairs indicates that we must pay attention to order for Cartesian products. In comparison, recall that the order of the elements in a set given in roster form does not matter. (See Example 5.20.)

### Example6.19.Sets versus ordered pairs.

We have the equality of sets $$\{1,2\} = \{2,1\}\text{.}$$ However, as ordered pairs, $$(1,2)\neq(2,1)\text{.}$$
Since the empty set $$\emptyset$$ does not contain any elements, there are no elements to be placed into the second component of the Cartesian product $$A\times \emptyset\text{.}$$ So, we have that $$A\times\emptyset=\emptyset$$ for any set $$A\text{.}$$ Similarly, $$\emptyset \times B = \emptyset$$ for any set $$B\text{.}$$
In Checkpoint 6.20 find all elements of a Cartesian product.

### Checkpoint6.20.Give Cartesian product in roster form.

Let $$A = \lbrace 5, 6, 7\rbrace$$ and let $$B = \lbrace 1, 2\rbrace\text{.}$$ Give the cartesion product of $$A$$ and $$B$$ in roster form:
$$A \times B = \lbrace$$$$\rbrace$$