Does a function with these properties exist?

Question

For $x_1, x_2, x_3 \in \mathbb{Z}^+$, does there exist a function $f(\cdot)$ defined on $\mathbb{Z}^+$, not necessarily continuous or differentiable, such that:

$$f(x_1) > f(x_2) \\ f(x_2) > f(x_3) \\ f(x_3) > f(x_1) $$

My immediate thought is that no such function exists, since a function can only be a many-to-one or one-to-one relation, and the above would require a one-to-many relation. If so, is there a more formal way of showing that the above is impossible? Or is it trivial to observe?

If I am wrong and such a function does exist, what is an example of one?

An interesting aside: https://en.wikipedia.org/wiki/Nontransitive_dice — Afiq Hatta, Jun 02 '20 at 07:55

score 2 · Accepted Answer · answered Dec 16 '18 at 12:29

2

A function has exactly one output.

If such a function exists, by transitivity, we would have $f(x_1) > f(x_1)$ which is a contradiction.

answered Dec 16 '18 at 12:29

Siong Thye Goh

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Does a function with these properties exist?

1 Answers1