Tutorial: Sets and set operations
This tutorial: Part B: Cartesian products and sets of outcomes
(This topic is also in Section 6.1 in Finite Mathematics or Section 7.1 in Finite Mathematics and Applied Calculus)
Cartesian products
Suppose you are at the dealership buying a new motorcycle, and you are down to deciding on an engine capacity and a color. The set of of capacities you are considering is  $V=\{$ 250, 350, 650, 750 $\}$ (in cc).
 $C = \{$ red, white, green $\}$.

$\displaystyle \begin{Bmatrix}(250, \text{red}), & (250, \text{white}), & (250, \text{green),}
\\ (350, \text{red}), & (350, \text{white}), & (350, \text{green}),
\\ (650, \text{red}), & (650, \text{white}), & (650, \text{green}),
\\ (750, \text{red}), & (750, \text{white}), & (750, \text{green})
\end{Bmatrix} \qquad$
We displayed the elements in four rows for convenience.

$\displaystyle V \times C = \begin{Bmatrix}(250, \text{red}), & (250, \text{white}), & (250, \text{green),}
\\ (350, \text{red}), & (350, \text{white}), & (350, \text{green}),
\\ (650, \text{red}), & (650, \text{white}), & (650, \text{green}),
\\ (750, \text{red}), & (750, \text{white}), & (750, \text{green})
\end{Bmatrix}$
Cartesian product
The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$:
The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$:
 $A \times B = \{ (a, b) \mid a \in A \text{ and } b \in B \}$.
Examples
1. \t !2! %%Let $A = \{a,b\}$ %%and $B = \{1,2,3\}$, Then \\ \t !2!
\t The elements of the cartesian product remind us of the way we represent points in the cartesian plane with two coordinates. Here, the "$x$coordinate" is an element of $A$, and the "$y$coordinate" is an element of $B$. So, we place the elements of $A$ along the $x$axis and the elements of $B$ along the $y$axis, and then look at the resulting coordinate grid, where each intersection point has coordinates that represent an element of $A \times B$. \\ \\ 2. \t %%If \\ \t \gap[40] $S = \{$%%H, %%T$\} \qquad$ \\ \t then \\ \t !2! \gap[40] $S \times S = \{$(%%H, %%H), (%%H, %%T), (%%T, %%H), (%%T, %%T)$\} \qquad$ \\ \\ 3. \t !3! Take R to be the set of all real numbers. Then \\ \\ \t !2! \gap[40] R × R $ = \{ (x, y) \mid x \text{ and } y $ real numbers $\}$, \\ \\ \t !3! which we recognize as a representation of the cartesian plane
Some for you
 $A \times B = \{(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)\}$.
\t The elements of the cartesian product remind us of the way we represent points in the cartesian plane with two coordinates. Here, the "$x$coordinate" is an element of $A$, and the "$y$coordinate" is an element of $B$. So, we place the elements of $A$ along the $x$axis and the elements of $B$ along the $y$axis, and then look at the resulting coordinate grid, where each intersection point has coordinates that represent an element of $A \times B$. \\ \\ 2. \t %%If \\ \t \gap[40] $S = \{$%%H, %%T$\} \qquad$ \\ \t then \\ \t !2! \gap[40] $S \times S = \{$(%%H, %%H), (%%H, %%T), (%%T, %%H), (%%T, %%T)$\} \qquad$ \\ \\ 3. \t !3! Take R to be the set of all real numbers. Then \\ \\ \t !2! \gap[40] R × R $ = \{ (x, y) \mid x \text{ and } y $ real numbers $\}$, \\ \\ \t !3! which we recognize as a representation of the cartesian plane
Sets of outcomes
We are often interested in the outcome of some kind of activity or "experiment". For instance: Toss a coin and observe which side faces up: There are two possible outcomes: %%heads (%%H) or %%tails (%%T), so the set of possible outcomes can be written as $S = \{\text{H},\text{T}\}$.
 Cast a die and observe the number facing up. We can represent the set of the six oucomes as $S = \{1, 2, 3, 4, 5, 6\}$.
Cartesian products and multistep experiments
If an experiment consists of two steps with individual sets of outcomes $A$ for the first step and $B$ for the second, then the set of outcomes for the twostep experiment is $A \times B$. Similarly, if an experiment consists of three steps with individual sets of outcomes $A, B$ and $C$ respectively for the individual steps, then the set of outcomes for the threestep experiment is $A \times B \times C$, the set of triples $(a,b,c)$ with $a \in A, b \in B$, and $c \in C$:
If an experiment consists of two steps with individual sets of outcomes $A$ for the first step and $B$ for the second, then the set of outcomes for the twostep experiment is $A \times B$. Similarly, if an experiment consists of three steps with individual sets of outcomes $A, B$ and $C$ respectively for the individual steps, then the set of outcomes for the threestep experiment is $A \times B \times C$, the set of triples $(a,b,c)$ with $a \in A, b \in B$, and $c \in C$:

$A \times B \times C = \{(a,b,c) \mid a \in A, b \in B, c \in C \}$.
Examples
\\ 1. \t %%If
\\ \t \gap[40] $S = \{$%%H, %%T$\} \qquad$
\\ \t !r! #[The set of outcomes of tossing a coin once][El conjunto de resultados al lanzar una moneda una vez]#
\\ \t then \\ \t !2! \gap[40] $S \times S = \{$(%%H, %%H), (%%H, %%T), (%%T, %%H), (%%T, %%T)$\} \qquad$
\\ \t !r! #[The set of outcomes of tossing a coin twice][El conjunto de resultados al lanzar una moneda dos veces]#
\\
\\ 2. \t %%If
\\ \t \gap[40] $S = \{1, 2, 3, 4, 5, 6\} \qquad$
\\ \t !r! #[The set of outcomes of rolling a die once][El conjunto de resultados al tirar un dado una vez]#
\\ \t then \\ \t !2! \gap[40] \t
\\ \t !r! The set of outcomes of rolling a die twice
\\ \t !3! Thus, for example, the outcome $(2, 3)$ represents a 2 followed by a 3 when you cast a die twice.
\\
\\ 3. \t Look again at the example of buying a motorcycle at the beginning of this tutorial.
\\ \t The decision as to which motorcycle to choose can be thought of as a twostep experiment:
Some for you
 Choose a capacity from the set $V=\{$ 250, 350, 650, 750 $\}$.
 Choose a color from the set $C = \{$ red, white, green $\}$.
Distinguishable and indistinguishable dice
We saw above that the set of outcomes when we throw a dice twice can be represented by a cartesian product

$\displaystyle \{1, 2, 3, 4, 5, 6\} \times \{1, 2, 3, 4, 5, 6\} = \begin{Bmatrix}
(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6),
\\ (2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6),
\\ (3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6),
\\ (4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6),
\\ (5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6),
\\ (6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6)
\end{Bmatrix}$.

Set of outcomes for a pair of indistinguishable dice $\displaystyle = \begin{Bmatrix}
(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6),
\\ \ & (2,2), & (2,3), & (2,4), & (2,5), & (2,6),
\\ \ & \ & (3,3), & (3,4), & (3,5), & (3,6),
\\ \ & \ & \ & (4,4), & (4,5), & (4,6),
\\ \ & \ & \ & \ & (5,5), & (5,6),
\\ \ & \ & \ & \ & \ & (6,6)
\end{Bmatrix}$.

Set of outcomes for a pair of indistinguishable coins $= \{$(%%H, %%H), (%%H, %%T), (%%T, %%T)$\}. \qquad$ #[We eliminated (%%T, %%H), as it is the same as (%%H, %%T).][Eliminaos (%%T, %%H), ya que es lo mismo que (%%H, %%T).]#
Now try the exercises in Section 6.1 in Finite Mathematics or Section 7.1 in Finite Mathematics and Applied Calculus.
Copyright © 2018 Stefan Waner and Steven R. Costenoble