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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
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, $a ⊙ b = b ⊙ a, \forall\ \text{a, b}\in\text{Q}$ 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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