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## Main Question or Discussion Point

A mathematical system S is defined as S={E,O,A}. A is the set of axioms describing the system. Is the definition of E considered an axiom? For example, if I want E={a,b}, then in the set A, do I write A={...,E={a,b},...}?

Also, is the definition of O an axiom? Say O={~,V} and then I define ~ to be a function from E to E such that ~={(a,b),(b,a)}. V is a binary operation on E such that V={((a,b),a),((b,a),a),((a,a),a),((b,b),b)}. Then can the property xVy=yVx be derived as a theorem based on the definition of V or must it be considered an axiom? Is the property ~(~(x))=x a theorem based on the properties of ~, or is it better to consider this an axiom? Which is more proper: to establish axioms describing the properties of the operation V and function ~ and from these properties determine their exact definition or to define the functions exactly and derive their properties (for this situation, the fact the V and ~ are of the exact form above is more important than the fact that they have the properties above)?

I hope my question is clear enough. I'm really not so sure what I'm asking myself.

Also, is the definition of O an axiom? Say O={~,V} and then I define ~ to be a function from E to E such that ~={(a,b),(b,a)}. V is a binary operation on E such that V={((a,b),a),((b,a),a),((a,a),a),((b,b),b)}. Then can the property xVy=yVx be derived as a theorem based on the definition of V or must it be considered an axiom? Is the property ~(~(x))=x a theorem based on the properties of ~, or is it better to consider this an axiom? Which is more proper: to establish axioms describing the properties of the operation V and function ~ and from these properties determine their exact definition or to define the functions exactly and derive their properties (for this situation, the fact the V and ~ are of the exact form above is more important than the fact that they have the properties above)?

I hope my question is clear enough. I'm really not so sure what I'm asking myself.