 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.
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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