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Question
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :
Find the identity element in A ?
Solution
\[\text{Let} E = (x, y) \text{be the identity element in A with respect to} \odot , \forall x \in R_0 \text{ & } y \in \text{R such that} \]
\[X \odot E = X = E \odot X, \forall X \in A\]
\[ \Rightarrow X \odot E = X \text{ and } E \odot X = X\]
\[ \Rightarrow \left( ax, bx + y \right) = \left( a, b \right) and \left( xa, ya + b \right) = \left( a, b \right)\]
\[\text{ Considering } \left( ax, bx + y \right) = \left( a, b \right)\]
\[ \Rightarrow ax = a \]
\[ \Rightarrow x = 1 \]
\[ \text{ & }bx + y = b\]
\[ \Rightarrow y = 0 \left[ \because x = 1 \right]\]
\[\text{Considering} \left( xa, ya + b \right) = \left( a, b \right)\] \[ \Rightarrow xa = a\]
\[ \Rightarrow x = 1\]
\[\text{ & } ya + b = b\]
\[ \Rightarrow y = 0 \left[ \because x = 1 \right]\]
\[ \therefore \left( 1, 0 \right) \text{is the identity element in A with respect to }\odot .\]
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