functional programming - How should I think about Scala's Product classes?

Question

The package "scala" has a number of classes named Product, Product1, Product2, and so on, up to Product22.

The descriptions of these classes are surely precise. For example:

Product4 is a cartesian product of 4 components

Precise, yes. Communicative? Not so much. I expect that this is the perfect wording for someone who already understands the sense of "cartesian product" being used here. For someone who doesn't, it sounds a bit circular. "Oh yes, well of course Product4 is the mumble product of 4 mumble-mumbles."

Please help me understand the correct functional-language viewpoint. What is the sense of "cartesian product" being used here? What do the Product classes' "projection" members indicate?

Answer 1

Everyone else has gone for the maths so I'll go for the silly answer just in case! You have a simple car which has a gearbox, a steering wheel, an accelerator and a number of passengers. These can each vary: which gear are you in, which way are you steering, is your foot "on the floor" etc. The gearbox, steering, accelerator etc are therefore variables and each has its own set of possible values.

The cartesian product of each of these sets is basically all possible states that your car can be in. So a few possible values are:

(gear,    steer,    accel,     pssngers)
--------|---------|----------|---------
(1st,     left,     foot down, none)
(neutral, straight, off,       the kids)

the size of the cartesian product is of course the product (multiplication) of the possibilities of each set. hence if you car has 5 gears (+ reverse + neutral), steering is left/straight/right, accelerator is on/off and up to 4 passengers, then there are 7 x 3 x 2 x 4 or 168 possible states.

This last fact is the reason that the cartesian product (named after Rene Descartes by the way) has the multiplication symbol x