"A surjective function is one that reaches all the elements of its codomain: some x does in fact reach every y. Another way of saying this is: for a surjective function, the range equals the entire codomain. You can see that this is impossible if the domain is smaller than the codomain, since there wouldn’t be enough x values to reach all the y values. If we added Pepsi and Barq’s Root Beer to our Y set, we would thereby eliminate the possibility of any surjective functions from X to Y, unless we also added wizards, of course. The function worksIn—with employees as the domain and departments as the codomain—is an example of a surjective function. One mapping of this function would be “worksIn(Sid) = Marketing,” indicating that Sid works in the Marketing department. Each employee works for one department, which makes it a function. But at least one employee works in every department, meaning there are no empty departments with no people in them, which makes it surjective. Surjective functions are sometimes called “onto” functions. Onto and surjective are exact synonyms."