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Question
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Find the invertible element in A ?
Solution
\[\text{ Let } \left(\text{ m, n }\right) \text{ be the inverse of } \left( a, b \right) \forall \left( a, b \right) \in A . \text{ Then }, \]
\[\left( \text{a, b} \right) * \left( \text{ m, n } \right) = \left( 1, 1 \right)\]
\[ \Rightarrow \left( \text{am, bn} \right) = \left( 1, 1 \right)\]
\[ \Rightarrow \text{am = 1 & bn }= 1\]
\[ \Rightarrow m = \frac{1}{a}\text{ & } n = \frac{1}{b}\]
\[\text{ Thus }, \left( \frac{1}{a}, \frac{1}{b} \right)\text{ is the inverse of } \left( a, b \right) \forall \left( a, b \right) \in A .\]
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