Definition:Cartesian Product/Class Theory

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This page is about Cartesian Cross Product in the context of Class Theory. For other uses, see Cross Product.

Definition

Let $A$ and $B$ be classes.


The cartesian product $A \times B$ of $A$ and $B$ is the class of ordered pairs $\tuple {x, y}$ with $x \in A$ and $y \in B$:

$A \times B = \set {\tuple {x, y}: x \in A \land y \in B}$


Thus:

$\forall p: \paren {p \in A \times B \iff \exists x: \exists y: x \in A \land y \in B \land p = \tuple {x, y} }$


$A \times B$ can be voiced $A$ cross $B$.


Also known as

Some authors call this the direct product of $S$ and $T$.

Some call it the cartesian product set, others just the product set.

Some authors use uppercase for the initial, that is: Cartesian product.

It is also known as the cross product of two sets, but this can be confused with other usages of this term.


The notation for the cartesian power of a set $S^n$ should not be confused with the notation used for the conjugate of a set.

Also beware not to confuse the name of the concept itself with that of the power set $\powerset S$ of $S$.


Examples

Product of Arbitrary Sets: 1

Let $S = \set {1, 2, 3}$.

Let $T = \set {a, b}$.

Then:

\(\ds S \times T\) \(=\) \(\ds \set {\tuple {1, a}, \tuple {1, b}, \tuple {2, a}, \tuple {2, b}, \tuple {3, a}, \tuple {3, b} }\)
\(\ds T \times S\) \(=\) \(\ds \set {\tuple {a, 1}, \tuple {a, 2}, \tuple {a, 3}, \tuple {b, 1}, \tuple {b, 2}, \tuple {b, 3} }\)


Product of Arbitrary Sets: 2

Let $V = \set {v_1, v_2}$.

Let $W = \set {w_1, w_2, w_3}$.

Then:

\(\ds V \times W\) \(=\) \(\ds \set {\tuple {v_1, w_1}, \tuple {v_1, w_2}, \tuple {v_1, w_3}, \tuple {v_2, w_1}, \tuple {v_2, w_2}, \tuple {v_2, w_3} }\)
\(\ds V \times V\) \(=\) \(\ds \set {\tuple {v_1, v_1}, \tuple {v_1, v_2}, \tuple {v_2, v_1}, \tuple {v_2, v_2} }\)


Also see

  • Results about Cartesian products can be found here.


Source of Name

This entry was named for René Descartes.


Historical Note

René Descartes is (rightly or wrongly) credited with inventing what is usually referred to as the Cartesian coordinate system of analytic geometry, in which points on (for example) the plane are identified with ordered pairs of real numbers.

Thus the plane is seen to be the Cartesian product of the set of real numbers $\R$ with itself.

Hence the reason for the name of the Cartesian product when used for general set or classes.


Sources

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