Question

Verify whether the operation * defined on Q by a * b = ab/4 is associative or not.

​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{On\;Q,\;a\,*\,b=\dfrac{ab}{4}}$

$\underline{\textbf{To verify:}}$

$\textsf{Whether * is associative or not}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Associative propterty:}}$

A binary operation * on S is said to be associative if

$\boxed{\mathsf{(a*b)*c=a*(b*c)}}$ for all $\mathsf{\;a,b,c\;\in\;S}$

$\mathsf{Let\;a,b,c\;\in\;Q}$

$\mathsf{Consider,}$

$\mathsf{(a*b)*c=\left(\dfrac{ab}{4}\right)*c}$

$\mathsf{(a*b)*c=\dfrac{\left(\dfrac{ab}{4}\right)c}{4}}$

$\mathsf{(a*b)*c=\dfrac{abc}{16}}$---------(1)

$\mathsf{a*(b*c)=a*\left(\dfrac{bc}{4}\right)}$

$\mathsf{a*(b*c)=\dfrac{a\left(\dfrac{bc}{4}\right)}{4}}$

$\mathsf{a*(b*c)=\dfrac{abc}{16}}$--------(2)

$\textsf{From (1) and (2), we get}$

$\mathsf{(a*b)*c=a*(b*c)}$

$\therefore\;\textsf{* is associative}$