Question

For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.
(i) On, $Z^{+}$ define $a\ast b=a-b$

Answer 1

Checking for binary.
Given: On $Z$, $a\ast b=a-b$
$\forall a,b\in Z,a-b\in Z$
So, ∗ is binary.
Checking for commutative.
Given: On $Z$, $a\ast b=a-b$
∗ is commutative if,
$a\ast b=b\ast a$
Now,
$a\ast b=b-a$
And,
$b\ast a=b-a$
Since, $a\ast b\ne b\ast a$
∗ is not commutative.
Checking for associative.
∗ is associative if,
$(a\ast b)\ast c=a\ast (a\ast c)$
$(a\ast b)\ast c=(a-c)\ast c$
$=(a-c)-c$
$=a-c-c$
$a\ast (b\ast c)=a\ast (b-c)$
$=a-(b-c$)
$=a-b+c$
Since $(a\ast b)\ast c\ne a\ast (b\ast c)$,
∗ is not an associative binary operation.