Question 1074717
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Let A={1,2,3,4,5}. 
(a) How many total functions f∶ A→A are there? 
(b) How many of the functions in (a) are one-to-one? 

Can someone please explain on how to start and solve these type of question? Thank you!
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<pre>
The short explanation is this:

(a)  Every function f: A --> A correlates some values of the set A to each element of the set A.
     Every function f: A --> A is such a correlation.

     Function f can correlate ANY of 5 values 1, 2, 3, 4 and/or 5 to the element "1" : FIVE options.

     Function f can correlate ANY of 5 values 1, 2, 3, 4 and/or 5 to the element "2" : FIVE options.

. . . and so on till 5.


   In all, there are {{{5^5}}} different functions.



(b)  One-to-one function in this context is simply a permutation ("without repetition").

    The number of permutations of 5 objects is 5! = 120 - very well known fact.
</pre>

Solved.