I have recently started learning about injective functions and can understand them to a basic level.
injective functions essentially equate to ∀x,y ∈ A f(x) = f(y) → x = y.
Assuming finite set S has the elements in a list: s1,s2,...,sn (where n = |S|).
Argue the list produces a topological ordering on S using f : S → Z
where f(s1) = 1, f(s2) = 2, which essentially generalises as f(sj) = j.
How would I argue that this is injective? Given the aforementioned rule would f(s1)=f(j)⟹1=j⟹j-1=0⟹s1=j be correct? Without 2 explicit variables, I'm unsure how to go about it.
Assuming a relation is refined R on S by aRb whenever f(a) ≤ f(b). Is R a total ordering on R? I would need to prove that it is reflexive, anti-symmetric, transitive and follows the Trichotomy Law. (∀x,y∈S xRy ∨ yRx ∨ x = y). I assume to prove it is anti-symmetric I would need to prove it is first injective? Similarly to the first question, I have a general direction, but no compass, help to understand these is appreciated.