Question

Let $R_0$ denote the set of all non-zero real numbers and let $A=R_0\times R_0$. If ${}^{\prime }{\ast }^{\prime }$ is a binary operation on $A$ defined by $(a,b)\ast (c,d)=(ac,bd)$ for all $(a,b)(c,d)\in A$.
Find the identity element in $A$.

Answer 1

Let $(e,e)$ be the identity element.

Then from the definition of the identity element we get,

$(a,b)\ast (e,e)=(e,e)\ast (a,b)=(a,b)$ for all $(a,b)\in R_0\times R_0$.

or, $(ae,be)=(a,b)$

Equating both sides we get,

$ae=a,be=b$

or, $e=1$. [ Since $a,b$ are non-zero elements]

So the identity element is $(1,1)$.