Cartesian product

In mathematics, given two sets X and Y, the Cartesian product (or direct product) of the two sets, written as X × Y is the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.

X × Y = { (x,y) | x in X and y in Y }

For example, if set X is the 13-element set {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} and set Y is the 4-element set {spades, hearts, diamonds, clubs}, then the Cartesian product of those two sets is the 52-element set { (A, spades), (K, spades), ... ,(2, spades), (A, hearts), ... , (3, clubs), (2, clubs) }. Another example is the 2-dimensional plane R × R where R is the set of real numbers - all points (x,y) where x and y are real numbers. Subsets of the Cartesian product are called binary relations.

The binary Cartesian product can be generalized to the n-ary Cartesian product over n sets X₁,... ,X_n:

X₁ × ... × X_n = { (x₁,... ,x_n) | x₁ in X₁ and ... and x_n in X_n }

Indeed, it can be identified to (X₁ × ... × X_n-1) × X_n. It is a set of n-tuples.

An example of this is the Euclidean 3-space R × R × R, with R again the set of real numbers.

The Cartesian product is named after Rene Descartes whose formulation of analytic geometry gave rise to this concept.

As an aid to its calculation, a table can be drawn up, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table by choosing the element of the set from the row and the column.

Children can be introduced to the Cartesian product by the familiar calendar:

• weeks as rows;
• weekdays as columns;
• a given day as a cell.

The Cartesian product can be used to graph mathematical properties, as in Graphing equivalence and Graphing the total product.

See also: Mathematics, Set theory, Group direct product