Question

Consider the binary operations $\ast :R\times R\to R$ and $o:R\times R\to R$ defined as $a\ast b=|a-b|$ and $aob=a$ for all $a,b\in R$. Show that ${}^{\prime }{\ast }^{\prime }$ is commutative but not associative, 'o' is associative but not commutative.

Answer 1

Two numbers are said to satisfy the commutative property if $p\ast q=q\ast p$

They satisfy the associative property if $p\ast (q\ast r)=(p\ast q)\ast r$

$a\ast b=|a-b|$, $b\ast a=|b-a|=|a-b|$

$(a\ast b)\ast c=|a-b|\ast c=||a-b|-c|...\left(1\right)$

$a\ast (b\ast c)=a\ast |b-c|=|a-|b-c||...\left(2\right)$

The expressions (1) and (2) might not be the same.

${\Rightarrow }^{\prime }{\ast }^{\prime }$ is commutative but not associative.

$aob=a,boa=b$

$\left(aob\right)oc=aoc=a,ao\left(boc\right)=aob=a$

${\Rightarrow }^{\prime }{o}^{\prime }$ is associative but not commutative.