Examine whether the operation *  defined on R by  a*b=ab+1  is  (i) a binary or not.  (ii) if a binary operation, is it associative or not ? 

 

 

 

 
 
 
 
 

Answers (1)

Given    a*b=ab+1\, \, \, \: \: \: \: \: \, \forall \: \: a,b\in \mathbb{R}

Being a binary operation, it should hold closure property.

a*b=\in \mathbb{R} \: \: \: \: \, \forall \: \: a,b\in \mathbb{R}

\therefore  It is  a*b=ab+1\: \: \: \: \: \: a,b\in \mathbb{R}

                        \Rightarrow real number only.

\therefore  It is binary.

To check associativity  \Rightarrow (a*b)*c=a*(b*c)

a*b=ab+1\: \: \: \: \: a,b\in\mathbb{R}

\therefore x,y,z\in\mathbb{R}

(x*y)*z=(xy+1)*z

                        =xyz+z+1            -- (1)

x*(y*z)=x*(yz+1)

                       =xyz+x+1            -- (2)

\therefore (1) and (2) are not equal

(x*y)*z\neq x*(y*z)

Hence structure is not associative.

 

 

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