Show that the operation x*y=(1/x) + (1/y) on Q, is well defined and commutative, but that it is not associative, nor does it have an identity.
Not really too sure how I would go about proving that the function is well defined on Q as if x and y are equal to zero, then they wouldn't be (at least I don't think they would be).
When I wanted to prove that the binary operation was commutative, I said:
x*y = (1/x) + (1/y) = (1/y) + (1/x) = y*x
I have no idea how to do the rest of the question as well. I tried:
Associativity: (x*y)z = x(y*z)
(1/x + 1/y)z = 1/x(1/y + z)
(1/x + 1/y)z = 1/x(1/y + z)
z/x + z/y = 1/xy + z/x
Untrue and therefore not associative.
Absolutely no idea how to find out if there is an identity or not. All I was able to do was define what an identity actually was.
Hint (for lacking an identity): Show that there exists no $e\in\boldsymbol Q$ such that $1\ast e=1$.