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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
'⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}$ then,

a ⊙ b = a2 + b2

= b2 + a2

= 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 ⊙ (b2 + c2)

= ab2 + (b2 + c2)2

= ab2 + b4 + c4 + 2b2c2

(a ⊙ b) ⊙ c = (a2 + b2) ⊙ c

= (a2 + b2)2 + c2

= a4 + b4 + 2a2b2 + c2

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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