Basic Abstact Algebra Proof

Hey guys, I am working on a problem in which the difficult part is finished, but I cannot wrap my head around how to prove the simple part.

I want to be able to use zero and the null set interchangably, but how in the world do I prove that they are equal??

To be more specific, I am trying to prove the following Axiom:

There is 0 in Z such that a+0 = a for each a in Z.

However, the values I am using for a are sets (such as {a,b,c}).

I just need to show that 0 = null set.

Thanks in advance for any help!