cross product
The cross product between two sets X and Y - symbolically written as X × Y - is another set Z = X × Y defined as the set of all possible prdered pairs (x,y) with x ∊ X and y ∊ Y.
In the special case where X and Y are discrete sets, such that X consists of the u elements x₁ x₂ ... xᵤ and Y consists of the v elements y₁ y₂ ... yᵥ, the cross product X×Y contains the following uv pairs:
(x₁,y₁) , (x₁,y₂) , ... , (x₁,yᵥ)
(x₂,y₁) , (x₂,y₂) , ... , (x₂,yᵥ)
...
(xᵤ,y₁) , (xᵤ,y₂) , ... , (xᵤ,yᵥ)
If the discrete sets X, Y represent coordinate axes - such as those used in objects of type Table -, the elements x₁ x₂ ... xᵤ in the indicated order form the (ordered) sequence of the coordinates of the axis X and the elements x₁ x₂ ... xᵤ form the (ordered) sequence of the coordinates of the axis Y.
Then the rowwise sequence of the above pairs constitute the (ordered) coordinates of a new axis Z = X × Y.
In a geometric sense, the cross product operation merges two axes into one, thus reducing a two-dimensional representation of data - such as a Table into a one-dimensional representation.
The definition of the cross product can be expanded on more than two sets, as follows:
X × Y × Z := (X × Y) × Z