What is the difference between concatenation and cross product of any 2 languages in Finite automata?

Question

I have come across many examples of concatenation and cross product but I still face difficulty in figuring out when to use either of them. Since both are combining the properties of any 2 languages I don’t know what makes both of them individually special. It’s confusing to pin point the clear differences between them. I hope if anyone can clarify this doubt of mine, I will be very grateful.

Answer 1

The concatenation of two languages is defined to be the cross product of the respective languages strings sets.

There is a nice table ("Closure properties of language families...") at Wikipedia in section "Operations on languages".

Answer 2

1. Cross Product

{cat, dog} X {1,2} = {(cat, 1), (cat, 2), (dog, 1), (dog, 2)}

2. Concatenation Product

{cat, dog} o {1,2} = {(cat1), (cat2), (dog1), (dog2)}