Properties of bijections

Question

If a bijection exists between set A={a1, a2, ...} and set B={b1, b2, ...} such that a1 maps to b1 and a2 maps to b2, etc., does this mean if we find a relationship R between a1 and b1 (i.e. f(b1) is always twice the size of f(a1)), then that relationship must also be true for a2 and b2?

Answer 1

No. Your $f$ could be $f(a_i) = i a_i$ if your sets are right. Your $f$ can be taken to set up a correspondence, like pairing up people with seats in a cinema. Many pairings are possible (just shuffle the people around).

Answer 2

The word you're looking for is compatibility ...

For a function $f$ and a relation $R$, we say that f respects R, or is compatible with it, iff $$\forall x,y :: x \ R \ y \implies (f \ x) \ R \ (f \ y)$$ ie related items have related images :)

Now if we have a bijection $f : A \cong B$ that is also compatible with a relation $R$, then we obtain the property you desire (... or not ...it's late where I am...)

Anyhow, hope this helps!