Associative and Commutative

Question

1. Determine which of the following operations are associative. Determine which are commutative.
   (a) Operation of * on Z (integer) defined by a∗b=a−b.
   (b) Operation of * on R (real numbers) defined by a∗b=a+b+ab.
   (c) Operation of * on Q (rational) defined by a∗b=a+b/5.
   (d) Operation of * on Z×Z defined by (a,b)∗(c,d)=(ad+bc, bd).
   (e) Operation of * on Q^∗ (=Q{0}) defined by a∗b=a/b.

Not asking for the answers but asking which steps should I take or is there a link I can use to understand how to do this. Anything helps!

Answer 1

To prove that the operation $*$ is commutative, you must show that $a*b=b*a$ for all $a$ and $b$.

If you find $a$ and $b$ such that $a*b\ne b*a,$ then you have shown that $*$ is not commutative.

To prove that $*$ is associative, you must show that $(a*b)*c=a*(b*c)$ for all $a,b, $ and $c$.

If you find $a, b, $ and $c$ such that $(a*b)*c\ne a*(b*c)$, then the operation $*$ is not associative.

Answer 2

All you need is to verify the definitions.

For whatever binary operation $*$, if you can prove that $a*b=b*a$ (using general elements, i.e. letters instead of numbers), probably using commutativity of some elementary operations like addition or multiplication or maximum, then $*$ is commutative.
To prove $*$ is not commutative, you only need to provide a specific example of $a, b$ (write numbers this time) such that $a*b\ne b*a$.

The same goes for associativity.