Let S={1,2,3,4}. Find the number of functions $\displaystyle f: S \to S$ such that $\displaystyle f(f(x)) = 1$ for all $\displaystyle x \in S$.

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- Oct 22nd 2008, 11:34 AMalexmahoneNumber of functions
Let S={1,2,3,4}. Find the number of functions $\displaystyle f: S \to S$ such that $\displaystyle f(f(x)) = 1$ for all $\displaystyle x \in S$.

- Oct 22nd 2008, 12:01 PMPlato
- Oct 22nd 2008, 12:13 PMPlato
- Oct 22nd 2008, 12:15 PMMoo
- Oct 22nd 2008, 12:25 PMPlato
And also:

$\displaystyle f:\left\{ {\left( {1,1} \right),\left( {2,1} \right),\left( {3,1} \right),\left( {4,2} \right)} \right\}$

$\displaystyle f:\left\{ {\left( {1,1} \right),\left( {2,1} \right),\left( {3,1} \right),\left( {4,3} \right)} \right\}$ - Oct 22nd 2008, 12:34 PMPlato
- Oct 22nd 2008, 02:28 PMPlato
There are $\displaystyle 4^4 = 256$ possible functions $\displaystyle \left\{ {1,2,3,4} \right\} \mapsto \left\{ {1,2,3,4} \right\}$.

Each of those functions is a set of four pairs: $\displaystyle f:\left\{ {\left( {1,\_} \right),\left( {2,\_} \right),\left( {3,\_} \right),\left( {4,\_} \right)} \right\}$.

But the pair $\displaystyle (1,1)$ must be there! (WHY?)

Suppose that $\displaystyle f(1) = 2$. Then in order for $\displaystyle f(f(1)) = 1 \Rightarrow \quad f(2) = 1$.

BUT $\displaystyle f(f(2)) = f(1) = 2 \ne 1$ so $\displaystyle f(1) \ne 2$. Thus only (1,1)

Another pair must have 1 as its second term.

What else is necessary?

If you work this out you will learn something about combinatorics - Oct 22nd 2008, 11:49 PMalexmahoneMy solution
The problem requires the following (obvious) conditions:

1) f(1)=1.

2) No element other than 1 can be mapped onto itself.

3) Atleast one of f(2), f(3), f(4) is equal to 1.

So f(2) can take 3 possible values:

__Case 1__: f(2)=1

If f(3)=1, f(4)=1 or 2 or 3

If f(3)=2, f(4)=1 or 2

If f(3)=4, f(4)=1

6 functions

__Case 2__: f(2)=3

f(3)=1 (forced)

f(4)=1 or 3

2 functions

__Case 3__: f(2)=4

f(4)=1 (forced)

f(3)=1 or 4

2 functions

Total=6+2+2=10 functions

Is that right? - Oct 23rd 2008, 02:42 AMPlato
That is what I got also.