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Suppose X is a finite set and we know that the number elements in X is $P^{2n}$ which $P$ is prime number and $n$ is an arbitrary natural number. Prove for every function $f:X \rightarrow X$ there are two metrics $d_1 , d_2$ on X such that for every $x_1,x_2 \in X$ we have $$d_1(x_1,x_2)

hint: I think $d_1$ is not important and you can define it easily and we have to focus on $d_2$.

hint: $d_1$ and $d_2$ are not discrete metric

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    What is a tow metric? – 2017-01-31
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    I think we can consider $d_1$ what ever we want but we have to prove there is $d_2$ – 2017-01-31
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    You write "hint: $d_1$ and $d_2$ are not discrete metric". Every metric on a finite set is discrete. – 2017-02-01

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