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Question
Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q ?
Solution
Commutativity :
\[\text{Let a}, b \in Q . \text{Then}, \]
\[a \odot b = a^2 + b^2 \]
\[ = b^2 + a^2 \]
\[ = b \odot a \]
\[\text{Therefore},\]
\[a \odot b = b \odot a, \forall a, b \in Q\]
Thus,
\[\odot\] is commutative on Q.
Associativity :
\[\text{Let } a, b, c \in Q . \text
{Then}, \]
\[a \odot \left( b \odot c \right) = a \odot \left( b^2 + c^2 \right)\]
\[ = a^2 + \left( b^2 + c^2 \right)^2 \]
\[ = a^2 + b^4 + c^4 + 2 b^2 c^2 \]
\[\left( a \odot b \right) \odot c = \left( a^2 + b^2 \right) \odot c\]
\[ = \left( a^2 + b^2 \right)^2 + c^2 \]
\[ = a^4 + b^4 + 2 a^2 b^2 + c^2 \]
\[\text{Therefore},\]
\[a \odot \left( b \odot c \right) \neq \left( a \odot b \right) \odot c\]
Thus, \[\odot\] is not associative on Q.
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