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 ?

 

Answer 1

\[\text{Let} E = (x, y) \text{be the identity element in A with respect to} \odot , \forall x \in R_0 \text{ & } y \in \text{R such that} \] 
\[X \odot E = X = E \odot X, \forall X \in A\] 
\[ \Rightarrow X \odot E = X \text{ and } E \odot X = X\] 
\[ \Rightarrow \left( ax, bx + y \right) = \left( a, b \right) and \left( xa, ya + b \right) = \left( a, b \right)\] 

\[\text{ Considering } \left( ax, bx + y \right) = \left( a, b \right)\] 
\[ \Rightarrow ax = a \] 
\[ \Rightarrow x = 1 \] 
\[ \text{ & }bx + y = b\] 
\[ \Rightarrow y = 0 \left[ \because x = 1 \right]\] 
\[\text{Considering} \left( xa, ya + b \right) = \left( a, b \right)\] \[ \Rightarrow xa = a\] 
\[ \Rightarrow x = 1\] 
\[\text{ & }  ya + b = b\] 
\[ \Rightarrow y = 0 \left[ \because x = 1 \right]\] 
\[ \therefore \left( 1, 0 \right) \text{is the identity element in A with respect to }\odot .\]