SOLUTION: Find all functions from {4, 5, 6} to {1, 2}.

First, let me explain you how to represent each such function.


You may represent it as a  (2x3)-matrix with 2 rows and 3 columns.


The lower row of such matrix is  (4, 5, 6).


The upper row of such matrix contains the VALUES of the function at the points 4, 5 and 6.



For example, you may have such matrix

    A1 = .


It represents the function, which has value of 1 at 4; value of 1 at 5;  and value of 2  at 6.



Another matrix might be

    A2 = .


It represents the function, which has value of 2 at 4; value of 2 at 5;  and value of 1  at 6.



Each matrix with the upper row consisting of ones and twos, represents some function of this family;

all lower rows of these matrices are the same  (4, 5, 6).



The functions differ by their upper rows: different functions have different upper rows, constructed as described.



The total number of such matrices and such functions is   = 8,

because you may have any of the two values 1 or 2 in the upper row in each its position, independently.



If you are enthusiastic, you may write all these 8 matrices, if you want to assure yourself that you do understand my explanation and the idea.



I will not do it, saving the gray substance of my mind.



I hope my description/explanation is clear enough to you.