Question

show that the binary operation * defined on set of all real numbers R by a*b=1,is both commutative and associative

Answer 1

$\textbf{Given:}$

In R, * is defined by a*b=1

$\textbf{To prove:}$

$\text{ * is commutative and associative}$

$\textbf{Solution:}$

$\textbf{Associative law:}$

$\bf\,a*(b*c)=(a*b)*c$

$\textbf{commutative law:}$

$\bf\,a*b=b*a$

$\textbf{Associative law:}$

$\text{Let a,b,c\,{\in}\,R}$Let a,b,c $\in$ R

$a*b=1$

$(a*b)*c=1*c=1$

$b*c=1$

$a*(b*c)=a*1=1$

$\implies\bf\,a*(b*c)=(a*b)*c$

$\therefore\textbf{ * is associative}$

$\textbf{Commutative law:}$

$\text{Let a,b\,{\in}\,R }$

$a*b=1$

$b*a=1$

$\implies\bf\,a*b=b*a$

$\therefore\textbf{ * is commutative}$