Writing One to One function in predicate logic.

Question

I'm a little confused on how to write a One to One function in symbolic form.

Initially, I was thinking it would be: $∀x,y∈Z, x != y → f(x)!= f(y)$, but my gut is telling me this is wrong. Any hints/tips?

Thanks,

Answer 1

For a function $f:Z\to Z'$ to be One to One means that there are no two different elements of $Z$ that get mapped to the same element of $Z'$. This formalizes to $$\forall x,y\in Z:x\neq y\Rightarrow f(x)\neq f(y)$$ Usually, the contrapositive, equivalent statement $$\forall x,y\in Z':f(x)=f(y)\Rightarrow x=y$$ is used.