This is also false for numerical product for a similar reason.
in ordinary language, if we never pick a first coordinate, it doesn't matter which second coordinate we were going to pick, it never gets chosen.
the same logic goes for the second coordinate: no matter which first coordinate we pick, if we never choose the second coordinate, we never get a pair.
picking an element of the empty set pretty much stops us dead in our tracks, it has no elements to pick.