Tutorial: Sets and set operations
This tutorial: Part B: Cartesian products and sets of outcomes
(This topic is also in 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 7.1 in Finite Mathematics and Applied Calculus.
Copyright © 2018 Stefan Waner and Steven R. Costenoble