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Let ∗ be a binary operation on the set Q of rational numbers as follows: a ∗ b = a 2 + b 2

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) a*b = a^2 + b^2

Find which of the binary operations are commutative and which are associative.

Answers (1)
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On the set Q ,the operation * is defines as  a*b = a^2 + b^2.It is observed that:

For a,b \in Q

a*b=a^{2}+b^{2}= b^{2}+a^{2}=b*a

\therefore      a*b=b*a 

Hence, the * operation is commutative.

It can be observed that

(1*2)*3 =(1^{2}+2^{2})*3 = 5*3 = 5^{2}+3^{2} = 25+9 =34

1*(2*3) =1*(2^{2}+3^{2}) = 1*13 = 1^{2}+13^{2} = 1+169 =170

  

(1*2)*3 \neq 1*(2*3)  for all 1,2,3 \in Q

 

The operation * is not associative.

   

 

 

 

 

 

 

 

 

 

 

  

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