TIFR 2014 Problem 30 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Topics in Algebra by I.N.Herstein. This book is very useful for the preparation of TIFR Entrance.
Also Visit: College Mathematics Program
How many maps $\phi: \mathbb{N} \cup {0} \to \mathbb{N} \cup {0}$ are there satisfying $\phi(ab)=\phi(a)+\phi(b)$ , for all $a,b\in \mathbb{N} \cup {0}$ ?
Take $n\in \mathbb{N} \cup {0}$.
By the given equation $\phi(n\times 0)=\phi(n)+\phi(0)$.
This means $\phi(0)=\phi(n)+\phi(0)$.
Oh! This means $\phi(n)=0$. $n\in \mathbb{N} \cup {0}$ was taken arbitrarily. So...
$\phi(n)=0$ for all $n\in \mathbb{N} \cup {0}$.
There is only one such map.
