Question

Let $*$ be a binary operation defined on $R$ by $a*b$ = $\frac {a+b}{4} \forall \, a,b \in R$ then the operation $*$ is

• A. Commutative and Associative
• B. Commutative but not Associative
• C. Associative but not Commutative
• D. Neither Associative nor Commutative

Answer 1

Answer: (B)

Solution: We have, $a * b=\frac{a+b}{4}$
For commutative, $a * b=b * a$
$\therefore \frac{a+b}{4}=\frac{b+a}{4}$
Hence, $*$ is commutative.
For associative, $a *(b * c)=(a * b) * c$
LHS $=a *\left(\frac{b+c}{4}\right)=\frac{a+\frac{b+c}{4}}{4}$
$=\frac{4 a+b+c}{16}$
$\text { RHS }=(a * b) * c=\left(\frac{a+b}{4}\right) * c $
$=\frac{\frac{a+b}{4}+c}{4}=\frac{a+b+4 c}{16}$
Since, $a *(b * c) \neq(a * b) * c$
So, ${ }^*$ is not associative.