Question

For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.

(iii) On $Q$, define $a\ast b=\frac{ab}{2}$

Answer 1

Checking for binary.
Given: On $Q$, $a\ast b=\frac{ab}{2}$
$\forall a,b\in Q,\frac{ab}{2}\in Q$
So, ∗ is binary.
Checking for commutative.
Given: On $Q$, $a\ast b=\frac{ab}{2}$
∗ is commutative if
$a\ast b=b\ast a$
Now,
$a\ast b=\frac{ab}{2}$
And,
$b\ast a=\frac{ba}{2}$
Since $a\ast b=b\ast a\forall a,b\in Q$,
∗ is commutative.
Checking for associative.
∗ is associative if,
$(a\ast b)\ast c=a\ast (b\ast c)$

$(a\ast b)\ast c=\left(\frac{ab}{2}\right)\ast c$
$\frac{\frac{ab}{2}\times c}{2}=\frac{abc}{4}$
$a\ast (b\ast c)=a\ast \left(\frac{bc}{2}\right)$
$\frac{a\times \frac{bc}{2}}{2}=\frac{abc}{4}$

Since, $(a\ast b)\ast c=a\ast (b\ast c)\forall a,b,c\in Q$
∗ is an associative binary operation.