Check the Commutativity and Associativity of the Following Binary Operations '⊙' On Q Defined By A ⊙ B = A2 + B2 For All A, B ∈ Q ? - Mathematics

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Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a² + b² for all a, b ∈ Q ?

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उत्तर

Commutativity :

$Let a, b \in Q . Then,$

$a ⊙ b = a^{2} + b^{2}$

$= b^{2} + a^{2}$

$= b ⊙ a$

$Therefore,$

$a ⊙ b = b ⊙ a, \forall a, b \in Q$

Thus,

$⊙$ is commutative on Q.

Associativity :

$Let a, b, c \in Q . Then,$

$a ⊙ (b ⊙ c) = a ⊙ (b^{2} + c^{2})$

$= a^{2} + {(b^{2} + c^{2})}^{2}$

$= a^{2} + b^{4} + c^{4} + 2 b^{2} c^{2}$

$(a ⊙ b) ⊙ c = (a^{2} + b^{2}) ⊙ c$

$= {(a^{2} + b^{2})}^{2} + c^{2}$

$= a^{4} + b^{4} + 2 a^{2} b^{2} + c^{2}$

$Therefore,$

$a ⊙ (b ⊙ c) \neq (a ⊙ b) ⊙ c$

Thus, $⊙$ is not associative on Q.

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Concept of Binary Operations

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Q 4.04 Q 4.03 Q 4.05

अध्याय 3: Binary Operations - Exercise 3.2 [पृष्ठ १२]

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अध्याय 3 Binary Operations
Exercise 3.2 | Q 4.04 | पृष्ठ १२

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Check the Commutativity and Associativity of the Following Binary Operations '⊙' On Q Defined By A ⊙ B = A2 + B2 For All A, B ∈ Q ? - Mathematics

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