Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
$'⊙\ '$ on $Q$ defined by $a ⊙ b = a^2 + b^2$ for all $a, b \in Q.$

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}$ then, $a ⊙ b = a^2 + b^2 = b^2 + a^2 = b ⊙ a$
Therefore $, a ⊙ b = b ⊙ a$, for all $\text{a, b}\in\text{Q}$ Thus, $⊙$ is commuatative on $Q$. Associativity : Let $\text{a, b, c}\in\text{Q.} a ⊙ (b ⊙ c) = a ⊙ (b^2 + c^2) = ab2 + (b^2 + c^2)^2 = ab^2 + b^4 + c^4 + 2b^2c^2 (a ⊙ b) ⊙ c$
$= (a^2 + b^2) ⊙ c = (a^2 + b^2)^2 + c^2 = a^4 + b^4 + 2a^2b^2 + c^2$
Therefore,$\text{a}\odot\text{b}\odot\text{c}\neq\text{a}\odot\text{b}\odot\text{c}$
Thus $, ⊙$ is not associative on $Q$.

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