1. ## binary operation

For each binary operation * defined, determine whether * is commutative and / or associative.

(a) on Z, define * by a*b = a-b
(b) on Q, define * by a*b=ab + 1
(c) on Z+, define * by a*b= 2^ab

The answers stated in my notes are
(a) not comm and not associative
(b) comm, not associative
(c) comm, not associative

Can someone care to explain to me the method to deduce the answers?
Thanks!

2. For commutative you have to check whether:

$a*b = b*a$
$a - b \ne b - a$

In this case, it's not closed under commutative.

For associative you have to check whether:

$(a*b)*c = a*(b*c)$
$(a - b)*c = a*(b - c)$
$a - b - c = a - (b - c)$
$a - b - c = a - b + c$

In this case, it's not closed under associative either.

Try the next two by yourself, and let me know how they come out.