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I have a homework problem that asks me to find a function from the set $\{1, 2, \ldots, 30\}$ to $\{1, 2, \ldots, 10\}$ that is a $3$-to-$1$ correspondence, I am confused on how to even derive a function from the sets that also are $3$-to-$1$ correspondence. In class we have had $k$-to-$1$ rule examples but never done $3$-to-$1$ or any $x$-to-$1$ problems so I am confused on how to apply it to my homework.

I will rewrite the problem for context:

Find a function from the set $\{1, 2, \ldots, 30\}$ to $\{1, 2, \ldots, 10\}$ that is a $3$-to-$1$ correspondence. (You may find that the division, ceiling or floor operations are useful).

Fred T
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Carrera
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  • $f(x)$ equals the least positive integer, such that $f(x) - x \equiv 0 \pmod{10}$. For example, $f(30) = f(20) = f(10) = 10, ~f(25) = f(15) = f(5) = 5.$ – user2661923 Nov 26 '22 at 03:05
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    A 3-to-1 function just takes three elements of the domain and maps them to one element of the range, (maybe, depending on the definition) once for each element of the range. One way is just to implement that. Choose three elements of the domain, say $1,9,15$ and map them to $2$. Choose three of the rest, say $12,19,27$ and map them to $7$. Keep going until you are done. The question is envisioning that you will have a nice formula for the function, but there is no requirement for that in the definition of a function. – Ross Millikan Nov 26 '22 at 03:25
  • To emphasize what RossMillikan is saying... there are MANY correct answers here. You are encouraged to be creative. For an added extra challenge, you might want to try asking exactly how many correct answers are possible. (the answer will be 25 digits long) – JMoravitz Nov 26 '22 at 04:07

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The 'obvious' way to do this would be to send $1,2,3$ to $1$, then $4,5,6$ to $2$ and so on, so $28,29,30$ goes to $10$. This would be $f(n) = \lceil \frac{n}{3} \rceil$.

Another (possibly neater) idea would be $1,11,21 \rightarrow 1$, then $2,12,22 \rightarrow 2$ etc. This would be $f(n) = n$ (mod $10$).

In fact, there are many different possible answers - most will not have a neat closed form like these, but any partition of $\{1, \ldots, 30\}$ into $10$ groups of $3$ can be such a function, by sending the first group of $3$ to $1$, the 2nd group to $2$ and so on

Fred T
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