Hello,
, this is according to the definition you're given.
It is pretty obvious that it is commutative, by commutativity of and .
As for associativity, you have to prove that . Just use the definition.
What is the identity, that is to say what is E such that ? . If and , you're done. The set E verifying this condition is
As for the inverse, you're looking for F such that .
If , then you're done.
So
For the last question, just rewrite the definition :
(the complement)