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Check the commutativity and associativity of each of the following binary operation:
\('\bigodot '\) on \(Q\) defined by \(a\bigodot b= a^{2}+b^{2}\) for all \(a,b \in Q\).


\(\bigodot\) is commutative if, \(a\bigodot b=b\bigodot a\)
\(\Rightarrow\) \(a\bigodot b=a^2+b^2\)
\(\Rightarrow\) \(b\bigodot a=b^2+a^2=a^2+b^2\)
\(\therefore\)\(a\bigodot b=b\bigodot a,\) \(\forall a,b\in Q\)
Thus, \(\bigodot\) is commutative on \(Q.\)
\(*\) is associative if,
\(\left(a\bigodot b\right)\bigodot c\) \(=\) \(a\bigodot(b\bigodot c)\)
\(\Rightarrow\)\(a\bigodot (b\bigodot c)=a\bigodot(b^2+c^2)\)

\(\Rightarrow\)\((a\bigodot b)\bigodot c=(a^2+b^2)\bigodot c\)
\(\therefore\) \(\left(a\bigodot b\right)\bigodot c\) \(\ne\) \(a\bigodot(b\bigodot c)\)
Thus, \(\bigodot\) is not associative on \(Q\).

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