Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators?
Two functions, f and g are equal to each other (denoted: f=g) if:
- They share the same domain.
- For every x in this domain, the values of f and g evaluated at x are equal.
In contrast, "two operators, O₁ and O₂ are said to be equivalent (denoted: O₁≣O₂) if for any function y to which O₁ and O₂ are each applicable, the functions O₁y and O₂y are equal."
Could someone please give me the rationale behind this distinction? Does this imply that operators are only equivalence relations while functions are equivalence relations and partial orders? And, if so, why are operators not also partial orders?
 Quoted from Samuel Goldberg's Introduction to Difference Equations.