Examine whether the operation * defined on R, the set of all real numbers, by a*b=\sqrt{a^2+b^2} is a binary operation or not, and if it is a binary operation, find whether it is associative or not.

 

 

 

 

 
 
 
 
 

Answers (1)

a\equiv R\: \: \: \: b\equiv R\: \: \: \: \: \: given\: \: a*b=\sqrt{a^2+b^2}

\Rightarrow a^2+b^2\equiv R

\Rightarrow \sqrt{a^2+b^2}\equiv R holds the closure property

Hence it is a Binary Equation .

Associative:

To Prove: a*b= \sqrt{a^2+b^2}\equiv R

\left (a*b \right )*c= a*(b*c)

\left ( \sqrt{a^2+b^2} \right )*c=a*(\sqrt{b^2+c^2})

\sqrt{a^2+b^2+c^2}=\sqrt{a^2+b^2+c^2}

\because LHS=RHS

It is associative

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