Cartesian Product
Definition
The Cartesian product of two sets \(A\) and \(B\), denoted \(A \times B\), is the set of all ordered pairs where the first element is from \(A\) and the second is from \(B\):
\[A \times B = \{(a, b) : a \in A, b \in B\}\]
Generalization
For \(n\) sets \(A_1, A_2, \ldots, A_n\):
\[A_1 \times A_2 \times \cdots \times A_n = \{(a_1, a_2, \ldots, a_n) : a_i \in A_i\}\]
Properties
Cardinality
\[|A \times B| = |A| \cdot |B|\]
Not Commutative
\[A \times B \neq B \times A \quad \text{(in general)}\]
Not Associative
\[(A \times B) \times C \neq A \times (B \times C)\]
(though they are naturally isomorphic)
Examples
Example 1
Let \(A = \{1, 2\}\) and \(B = \{x, y\}\).
\[A \times B = \{(1, x), (1, y), (2, x), (2, y)\}\]
Example 2: \(\mathbb{R}^2\)
\[\mathbb{R} \times \mathbb{R} = \mathbb{R}^2\]
The Euclidean plane.