Relationship between cartesian product and cross product?

Question

Is there any relation between cartesian product and cross product? Or is it just the same symbol?

Answer 1

It is the same symbol used for different concepts; unfortunately there are too many "products" in mathematics.

The "extension" of the product operation from numbers to vectors, gives rise to two different operations; see :

• Josiah Willard Gibbs (1839 – 1903), Vector Analysis (textbook by E.B.Wilson, first published in 1901 and based on the lectures that J.W.Gibbs had delivered at Yale University); see pages 55 and 60 :

to introduce two new combinations of vectors. These will be called products because they obey the fundamental law of products; i.e., the distributive law [...].

Definition : The direct product of two vectors $A$ and $B$ [denoted : $A \cdot B$] is the scalar quantity obtained by multiplying the product of the magnitudes of the vectors by the cosine of the angle between them. [...]

Definition : The skew product of the vector $A$ into the vector $B$ [denoted : $A \times B$] is the vector quantity $C$ whose direction is the normal upon that side of the plane of $A$ and $B$ on which rotation from $A$ to $B$ through an angle of less than one hundred and eighty degrees appears positive or counterclockwise; and whose magnitude is obtained by multiplying the product of the magnitudes of $A$ and $B$ by the sine of the angle from $A$ to $B$.

---

Regarding the algebra of sets operations, terminology has changed over time.

The operations of sets union : $A \cup B$ and set intersection : $A \cap B$ was originally named sum and product, respectively, deriving their name from the Algebra of Logic of Boole.

Probably, union is found in 1912 in James Pierpoint's Lectures on the theory of functions of real variables (Vol. 2, p. 22).

We can see this in the first modern set theory textbook :

• Felix Hausdorff, Grundzüge der Mengenlehre (1914, Engl.transl. of the 3rd German ed., 1937), page 18 and 23 :

If $A, B$, are two sets, then by their sum [German 1914, page 5 : Summe], or union,

$$S = A + B$$

we mean the set of all elements that belongs to $A$ or $В$ (or both) ; by their intersection [German 1914 : Durchschnitt]

$$D = AB$$

we mean the set of all elements that belong to both $A$ and $B$. [...]

It remains to define the product of sets. Let us form from two sets $A$ and $В$ the set $Ρ$ of ordered pairs $p=(a,b)$, where $a$ runs through all the elements of $A$ and $b$ runs through all the elements of $B$, [...]. we define the ordered pair of sets to be the set of ordered pairs of elements [German 1914, page 34 : Menge der Paare]. [...] Thus

$$P = (A,B)$$

is the product of the two sets.

Cartesian product entered circulation in the 1930s. Previously product (Produkt) was the established term (see, e.g. F.Hausdorff, Grundzüge der Mengenlehre (1914, p. 37)). Kuratowski wrote produit for intersection and produit cartésien for the former product.

See Kazimierz Kuratowski, Topologie , Vol.I (1933), page 12 :

Définitions. Le produit cartésien (qu'il ne faut, pas confondre avec le produit = partie commune) des ensembles $A$ et $B$ est l'ensemble $A \times B$ composé de tous les couples ordonnés $a,b$ où $a \in A$ et $b \in B$.

See also :

• Waclaw Sierpinski, Algèbre des ensembles (1951), page 62, we have :

Nous appelons somme (ou réunion) des ensembles $A$ et $B$ et nous désignons par $A+B$ l'ensemble formé [...]. Nous appelons produit (ou intersection) des ensembles $A$ et $B$ et nous désignons par $A \cdot B$ l'ensemble [...]

and page 108 :

Par produit cartésien (ou combinatoire) de deux ensembles $A$ et $B$, on comprend l'ensemble de toutes les paires ordonnées $(x,y)$, où $x \in A$ et $y \in B$ : un tel ensemble est désigné par $A \times B$.

The final step was to remove the reference to "product" for intersection, so to avoid confusion.