C.3 The Cartesian product

One of the most fundamental ways to construct new sets is the Cartesian product.

Definition 110. (The Cartesian product)

Let A and B be two sets. Their Cartesian product A ×B is defined by

The elements of the product are called tuples. Note that this operation is not associative nor commutative!

To see this, consider that, for example,

and

The Cartesian product for an arbitrary number of sets is defined with a recursive definition, just like we did with the union and intersection. So, if A₁,A₂,…,A_n are sets, then

Here, the elements are tuples of tuples of tuples of…, but to avoid writing an excessive number of parentheses, we can abbreviate it as (a₁,…,a_n). When the operands are the same, we usually write Aⁿ instead of A ×⋅⋅⋅×A.

One of the most common examples is the Cartesian plane, which you probably have seen before.

Figure C.3: The Cartesian plane

To give a machine-learning...