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.

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## PROBLEM:

How many maps \(\phi: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}\) are there satisfying \(\phi(ab)=\phi(a)+\phi(b)\) , for all \(a,b\in \mathbb{N} \cup \{0\}\) ?

## Discussion:

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.

## HELPDESK

**What is this topic:**Algebra**What are some of the associated concept:**Number of Function**Book Suggestions:**Topics in Algebra by I.N.Herstein