Here is a basic, though very important, example: Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of constructing a segment congruent to an assigned segment”. His second axiom reads: "if two segments are congruent to a third one they are congruent to each other." Then he goes to prove the standard properties of equivalence relations follow from the axioms: Since congruence or equality is introduced in geometry only through these axioms, it is by no means obvious that every segment is congruent to itself. However, this fact follows from the first two axioms on congruence if the segment AB is constructed on a ray so that it is congruent, say, to A'B' and Axiom III, 2 is applied to the congruences AB ≡ A'B', AB ≡ A'B'. On the basis of this the symmetry and the transitivity of segment congruence can be established by an application of Axiom III, 2.

Now it is clear how we may define the general notion of equivalence relation. This "Hilbertian" defintion has at least two advantages: first, It avoids (at the start) the somehow non-intutive property of reflexivity; second, and more importantly, it is in more harmony with the standard way of defining an equivalence class consisting of everything equivalent to a focal element.

Here are my sub-questions: Was the standard definition of equivalence relation just more lucky? Or, was there a rational choice involved? Do you know any other definition that has been preferred over the other? If yes, what was the reason? Do you have a personal example of such "choice"? If yes, what is the reason of your preference? Do you have an "instructional" defintion of a certain concept that you prefer it over the "standard" definition of the same concept when teaching? if yes,...

simplerthe idea becomes (and therefore easier to communicate), rather than becoming too big or too hard. Definitions are awkward because language is not always perfectly suited to express a mathematical idea concisely. $\endgroup$ – Paul Apr 19 '13 at 13:31