First, a bit of context. About a quarter of an hour ago I came across one of those "Internet math puzzles" on Facebook that stated:
If 1 = 5, 2 = 10, 3 = 15, and 4 = 20, then 5 = ?
The answer was supposed to be 1, as we had already stated. But of course, assuming the rule is x = 5x, then 5 is equal to both 1 and 25, as well as 125, 625, 3125, etc.
This got me thinking, can we define a relation like the one the question asks for so that 5 isn't mapped to 25, but x is still mapped to 5x in general? What I got was the following:
$x = \left\{ \begin{matrix} 5x & \text{the prime factorization of } x \text{ has an even power of 5} \\ x/5 & \text{the prime factorization of } x \text{ has an odd power of 5} \end{matrix} \right.$
I'm pretty sure this is a bijection. But if it is, I don't know if there's a special name for bijections like this, that map pairs of values to each other.
Basically, what's the name for the type of bijection $f: D \to D$, such that for any $a, b \in D$, if $f(a) = b$, then $f(b) = a$, and $a \neq b$?