The Cartesian product is an essential notion to know in set theory. Let's discover its definition together.

Definition of the Cartesian product

We call Cartesian product of two sets A and B (it is denoted A x B) the set of couples (a;b) where a is an element of A and b an element of B. Mathematically, this is written:

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

When A and B are finite, the cardinality of A x B is the product of the cardinality of A and the cardinality of B. So if the cardinality of A is p, that of B is q then the cardinality of A x B is pq.

\#(A\times B) = \#A \times \#B

Here is some additional information:

• The Cartesian product is not commutative, E x F is generally different from F x E, they are also different when F is different from E
• If F = E, we can replace the notation E x E by E²
• We can define a Cartesian product with n terms:

A_1 \times \ldots \times A_n = \{ (a_1,\ldots,a_n), a_1 \in A_1, \ldots, a_n \in A_n \}

• And if all the A's_i are equal to A, we also have the following notation: A₁ x … x A_n = Aⁿ which is much more condensed.

Examples

• If A = {a,b} and B = {1,2,3} then the set A x B resulting from the Cartesian product is made up of the following 6 elements: A x B = {(a,1); (a,2); (a,3); (b,1); (b,2); (b,3)}. Note that B x A is quite different from A x B and equals {(1,a); (2,a); (3,a); (1,b); (2,b); (3,b)}
• If A and B are equal to the set of real numbers, we define

\mathbb{R}^2

by

\mathbb{R}^2 = \{(a,b), a \in \R, b\in \R\}

The Cartesian Product in SQL

To make the connection with the SQL, the joint cross join corresponds exactly to the Cartesian product. If we have a first table of k rows and a second table of l rows then the cross join between these two tables gives a table of kxl rows corresponding to the Cartesian product of the 2 rows.