TIFR 2014 Problem 30 Solution - Number of Maps

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