If we perform an operation on ordered pair from set A={1,2,3,4}, we can built a table like this:
But, since set A is finite and there is only one element "1", "2" etc., why we can double elements in the table? An operation cannot be perform like this: 1*1, because we have only one "1". In other words, my question is: what we choose from given set A: an ordered pair OR each element of an ordered pair?
An operation on $A$ is a mapping from $A\times A$ to $A$. Since $(1,1)$ is an element of $A\times A$, $1*1$ is defined.
There is nothing in the definition of an operation that would forbid us from "doubling" elements in the table. That, in essence, is the only answer we can give to the question