Cartesian Product

Just as we can have a set of sets, we can also have a set of sequences.

The Cartesian product of sets $A$ and $B$ is a set containing all the possible ordered pairs (points) of the form $(a, b)$, in which $a \in A$ and $b \in B$. In other words, it is the set of sequences of 2 elements, such that the first coordinate is an element in the first set ($A$), and the 2nd coordinate is an element in the second set ($B$). The notation of the Cartesian product is $A \times B$ (

$A \times B$

). Examples:

• For $A = \{0, 1, 2\}$ and $B = \{8, 9\}$, we have $A \times B = \;$$\{(0, 8), (0, 9), (1, 8), \,$$(1, 9), (2, 8), (2, 9)\}$.
• For $S = \{1, 2, 3\}$, we have $S \times S = \;$$\{(1, 1), (1, 2), (1, 3), \,$$(2, 1), (2, 2), (2, 3), \,$$(3, 1), (3, 2), (3, 3)\}$.

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If $\boldsymbol{|A| = n}$ and $\boldsymbol{|B| = m}$, then how much is $\boldsymbol{|A \times B|}$ equal to? Can you spot a pattern in the examples above?