In the problems involving two algebraic systems, for eg.,$\langle S,*\rangle$ and $\langle P,\bigoplus\rangle$ where the sets $S=\{a,b,c\}$ and $P=\{1,2,3\}$. Here we have to check whether they both are isomorphic or not. While solving, they take values as $g(a)=3$, $g(b)=1$ and $g(c)=2$ and prove the systems as isomorphic. If I try other combination of values, it doesn't satisfy isomorphism. Then, on what basis these values are chosen(a=3,b=1,c=2)?
Kindly check out this Pg. 234 for the definitions of the operations.