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},...}?(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Math system

**Physics Forums | Science Articles, Homework Help, Discussion**