Properties of Operations problem

Question

Please help I can't figure out this problem

Let Z denote the set of integers: { …, -3, -2, -1, 0, 1, 2, 3, …}. Define an operation  on Z by x  y = 2(x + y).

a. Is Z closed under  ? Explain. b. Is  associative? Explain. c. Is there an identity? Explain. d. Is there inverses? Explain.

Answer 1

a. If $x,y\in Z$, then $xy=2(x+y)\in Z$, so it's closed.

b. $2(2(x+y)+z)$ does not generally equal $2(x+2(y+z))$, so it's not associative.

c. There is no $x$ such that, for all $y$, $2(x+y)=y$, so there's no identity.

d. There are no inverses, because there is no identity.