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 ?

Answer 1

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