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Discrete Mathematical Structures

Dan Yasaki, Sebastian Pauli

Subsection 2.1.5 Cartesian products

Definition 2.1.5.1.

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 \(b \in B\text{.}\) That is,
\begin{equation*} A \times B = \set{(a,b) \given a \in A\land b \in B}. \end{equation*}
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Example 2.1.5.2. A Cartesian product.

Let \(A = \set{0,1}\text{,}\) and let \(B = \set{4,5,6}\text{.}\) Then the Cartesian product \(A \times B\) is the set of all ordered pairs \((a,b)\text{,}\) where \(a\) is an element from \(A\) and \(b\) is an element of \(B\text{.}\) Thus
\begin{equation*} A \times B = \set{(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6)}. \end{equation*}
Analogously, the Cartesian product \(B \times A\) is the set of all ordered pairs \((b,a)\text{,}\) where \(b\) is an element of \(B\) and \(a\) is an element of \(A\text{.}\) Thus
\begin{equation*} B \times A = \set{(4, 0), (4, 1), (5, 0), (5, 1), (6, 0), (6,1)}. \end{equation*}
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More generally, we can define the Cartesian product of more than two set.
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Definition 2.1.5.4.

The Cartesian product of \(n\) sets \(A_1, A_2, \dots, A_n\text{,}\) denoted \(A_1 \times A_2 \times \dots \times A_n\) is the set of ordered \(n\)-tuples \((a_1, a_2, \dots, a_n)\) where \(a_i \in A_i\) for \(i = 1, 2, \dots, n\text{.}\) That is,
\begin{equation*} A_1 \times A_2 \times \dots \times A_n = \set{(a_1, a_2, \dots, a_n) \given a_i \in A_i\text{ for }i = 1, 2, \dots, n}. \end{equation*}
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Definition 2.1.5.5.

Let \(A\) be a set. We use the notation \(A^2\) to denote the Cartesian product \(A \times A\text{.}\) Similarly we write \(A^3\) for the Cartesian product \(A \times A \times A\text{,}\) and so on.
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Example 2.1.5.6. The Cartesian plane.

\(\RR^2\) is the familiar Cartesian plane.
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