Cartesian Product

Definition: For two sets $A$ and $B$, the cartesian product $A \times B = \{ (a, b) : a \in A \: \mathrm{and} \: b \in B \}$ is a set containing all ordered pairs $(a, b)$ where $a$ is an element of $A$ and $b$ is an element of $B$.

For example, consider the set $A = \{ 1, 2, 3 \}$ and $B = \{ 4, 5, 6 \}$. The cartesian product $A \times B = \{ (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6) \}$. Note that elements in $A \times B$ are ordered pairs $(a, b)$. Therefore, the element $(4, 1) \not \in A \times B$ since $4 \not \in A$ and $1 \not \in B$.

Now consider two sets $A$ and $B$. The cartesian product $A \times B$ can be represented on the plane if we know what sort of elements are in $A$ and $B$. For example, consider the set $A = \{ x : x \in \mathbb{R} \: \mathrm{and} \: 1 ≤ x ≤ 2 \}$ and the set $B = \{ x : x \in \mathbb{R} \: \mathrm{and} \: 3 ≤ x ≤ 4 \}$. The cartesian product $A \times B$ can be visualized as follows: