I need to figure out whether these binary operations are commutative or associative. And then whether a unity exists (but I don't know what that means).
- M=$\mathbb{Z}$; a*b=a-b
- M=$\mathbb{Q}$; a*b=$\frac{1}{2}$ab
I understand commutative property but not associative for these.
Commutative means $a*b = b*a$.
Associative means $a*(b*c) = (a*b)*c$.
A unity element $e$ is one such that $a*e = a$ for all $a$ (this is a right unity - a left unity is defined similarly).
So, put these definitions into your operator definitions and see which holds and which do not. For the unity, see what properties the unity element must have.