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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 identity element in A ?
Solution
\[\text{Let} \left( x, y \right) \text{be the identity element in A} \forall \left( x, y \right) \in \text{ A . Then }, \]
\[\left( a, b \right) * \left( x, y \right) = \left( a, b \right) = \left( x, y \right) * \left( a, b \right) \]
\[ \Rightarrow \left( a, b \right) * \left( x, y \right) = \left( a, b \right) \text{ and } \left( x, y \right) * \left( a, b \right) = \left( a, b \right)\]
\[ \Rightarrow \left( ax, by \right) = \left( a, b \right) \text{ and } \left( xa, yb \right) = \left( a, b \right)\]
\[ \Rightarrow x = 1 \text{ and } y = 1 \]
\[\text{Thus }, \left( 1, 1 \right) \text{is the identity element of A } . \]
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