Question:
A=R X R and * be the binary operation on A defined by (a,b)*(c,d)=(a c,b d).Find identity element for * on A.
Answer:
Given (a, b)*(c, d) = (a+c, b+d)
Let an element e = (e1 ,e2 ) ∈ A will be an identity element for thr operation *.
Now a*e = a = e*a for all a = (a1 ,a2 ) ∈ A
i.e. (a1 + e1 ,a2 + e2 ) = (a1 ,a2 ) = (e1 + a1 , e2 + a2 )
Which is not true for any element in A
So, the operation * does not have any identity element.
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