I think you need to clean up your work a bit. You're trying to show that

Incidentally, both sides of this equation represent the symmetric difference of and .

What you need to show is that each side is a subset of the other. The statement you wrote

is incorrect. The correct statement would beWe need to show is contained in

and is a superset of

For the forward direction, you start out correctly:We need to show is contained in

and is contained in

But then you say this:Let be an element of .

That is incorrect. It should be this:By definition of union, is in or is not in .

Can you continue from here, or are you still stuck?By definition of union, is in or is in .