Symmetric Difference is Associative

Theorem
Symmetric difference is associative:


 * $$R * \left({S * T}\right) = \left({R * S}\right) * T$$

Proof 1
We can directly expand the expressions for $$R * \left({S * T}\right)$$ and $$\left({R * S}\right) * T$$, and see that they come to the same thing.

Expanding the RHS:

$$ $$ $$ $$ $$

Expanding the LHS:

$$ $$ $$ $$ $$

Thus we see that (by applying the fact that Union is Commutative) the LHS and RHS are the same, and the result is proved.

Proof 2
There is another way of doing this.

Expanding the RHS:

$$ $$ $$ $$ $$

Expanding the LHS:

$$ $$ $$ $$ $$

Similarly, we see that (by applying the fact that Intersection is Commutative) the LHS and RHS are the same, and the result is proved.

Comment
This illustrates that you can express the symmetric difference of three sets as the union of four intersections (which seems more intuitively obvious) as well as the intersection of four unions (which is not quite so obvious).