Browse Questions

# (i) Is the binary operation defined on set N, given by $a*b = \large\frac{a+b}{2}$ for all a,b $\in$ N. commutative? (ii) Is the above binary operation associative?

Toolbox:
• An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
• An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Step 1:
The binary operation is given by
$a \ast b=\large\frac{a+b}{2}$$\;(a,b\in N) \therefore b\ast a=\large\frac{b+a}{2} for all a,b\in N Hence it is commutative. Step 2: a\ast (b\ast c)=(a\ast b)\ast c LHS : a\ast \big(\large\frac{b+c}{2}\big)=\large\frac{a+\Large\frac{b+c}{2}}{2} \Rightarrow \large\frac{2a+b+c}{4} RHS : (a\ast b)\ast c=\large\frac{a+b}{2}$$\ast c$
$\Rightarrow\large\frac {\Large\frac{a+b}{2}+c}{2}$
$\Rightarrow \large\frac{a+b+2c}{4}$
Clearly $LHS\neq RHS$
So it is not associative.