Let A be the set of odd numbers: 1, 3, 5, 7, etc. and B be the set of squares of the natural numbers,
The intersection of A and B is the set of all odd numbered squares: 1, 9, 25, etc.
B' is the set of all numbers that are not the set of squares of the natural numbers.
The intersection of B' and A is the set of all odd numbers except for the odd numbered squares: 3, 5, 7, 11, ...
(B'^A)v(A^B) combines 1, 9, 25, ... with 3, 5, 7, 11, ... thereby restoring the extracted elements of A.
Therefore A=(B'^A)v(A^B).
This applies to all sets A and B.