- Is (A) natrual join (A) = A? Yes
- Is (A) natrual join (empty set) = A? No
Here is a working demo.
Since natural join compares all columns in the two tables that have the same column names and return only one column for same-name columns, A join A will result in A. For point #2, join will not match any rows so the result will be empty table (not A).
Note: The Cartesian product of A and the empty set is the empty set
Also, check this Why is the Cartesian product of a set A and empty set an empty set?