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(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?

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  • 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}$$\;(a,b\in N)$
$\therefore b\ast a=\large\frac{b+a}{2}$ for all $a,b\in N$
Hence it is commutative.
Step 2:
$a\ast (b\ast c)=(a\ast b)\ast c$
$a\ast \big(\large\frac{b+c}{2}\big)=\large\frac{a+\Large\frac{b+c}{2}}{2}$
$\Rightarrow \large\frac{2a+b+c}{4}$
$(a\ast b)\ast c=\large\frac{a+b}{2}$$\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.
answered Sep 26, 2013 by sreemathi.v

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