**Definition 3.2.29**
Let

and

be any two sets. The symmetric difference of

and

is the set denoted by

and who consists of all the
elements that belong to exactly one of the sets

and

, and not
to both, i.e.

*Proof*.
We can write down a proof using truth tables; as this is exactly the proof of the associative law of the ``exclusive or'' from Chapter

2, we are done.
Anyway, we wish at least to outline a proof using properties of
operations on sets. On the one hand, we have: (the reader will fill
the missing rows):

On the other hand, we have:

Whence the required equality.