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)