How do we show that the set of connectives is not complete?

I'm not sure how to prove it is or it is not complete. But I know how to test for adequacy. And I know the set of standard connectives is adequate , so a set of connectives is adequate if we can express all the standard connectives in terms of this set. There is a test for adequacy of connctives. But if prove that the set is not adequate, how do we know if it failed completeness?