I need to prove this: X ∆ X = ∅

Prove: X ∆ X = ∅ ?

Theorem: X ∩ X = X and X ∪ X = X by the idempotence of the operations of set union and set intersection.

Definition:
X ∆ X = {a | a ∈ X ∪ X and a ∉ X ∩ X}
X ∆ X = {a | a ∈ X ∪ X } ∩ {a | a ∉ X ∩ X }

However, since we have stated that X ∩ X = X ∪ X = X

X ∆ X = {a | a ∈ X} ∩ {a | a ∈ X'}
X ∆ X = X ∩ X'
X ∆ X = ∅

I'm a bit rusty on my proofs, but you get the idea. You can optionally choose to skip the whole "a such that a is an element of..." I simply do this to begin with the proposition that perhaps there exists some element a in these sets.

When ∆x is a very small quantity, ∆x --> 0
So (any finite quantity)*(∆x) --> 0, and x being a finite quantity ,
x*(∆x) --> 0

Braaaaaaiiiiins!