Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $\to$ S
such that f(m . n) = f(m) . f(n) for every m, n $\in$ S and m . n $\in$ S is equal to _____________.
Answer (integer)
490
Solution
F(mn) = f(m) . f(n)<br><br>Put m = 1 f(n) = f(1) . f(n) $\Rightarrow$ f(1) = 1<br><br>Put m = n = 2<br><br>$$f(4) = f(2).f(2)\left\{ \matrix{
f(2) = 1 \Rightarrow f(4) = 1 \hfill \cr
or \hfill \cr
f(2) = 2 \Rightarrow f(4) = 4 \hfill \cr} \right.$$<br><br>Put m = 2, n = 3<br><br>$$f(6) = f(2).f(3)\left\{ \matrix{
when\,f(2) = 1 \hfill \cr
f(3) = 1\,to\,7 \hfill \cr
\hfill \cr
f(2) = 2 \hfill \cr
f(3) = 1\,or\,2\,or\,3 \hfill \cr} \right.$$<br><br>f(5), f(7) can take any value <br><br>Total = (1 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7) + (1 $\times$ 1 $\times$ 3 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7)<br><br>= 490
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
This question is part of PrepWiser's free JEE Main question bank. 195 more solved questions on Sets, Relations and Functions are available — start with the harder ones if your accuracy is >70%.