Cartesian Product

Definition

The Cartesian product of two sets $A$ and $B$ , denoted $A \times B$ , is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$ .

$A \times B = {(a, b) ∣ a \in A, b \in B}$

For $n$ sets $A_{1}, A_{2}, \dots, A_{n}$ , the Cartesian product is:

$A_{1} \times A_{2} \times \dots \times A_{n} = {(a_{1}, a_{2}, \dots, a_{n}) ∣ a_{i} \in A_{i} for all i}$

If either $A$ or $B$ is empty, then $A \times B = \emptyset$
$∣ A \times B ∣ = ∣ A ∣ \cdot ∣ B ∣$ when $A$ and $B$ are finite
The Cartesian product is not commutative: $A \times B \neq = B \times A$ in general
The Cartesian product is associative up to isomorphism: $(A \times B) \times C ≅ A \times (B \times C)$

If $A = {1, 2}$ and $B = {x, y}$ , then:

$A \times B = {(1, x), (1, y), (2, x), (2, y)}$