Prove that there is a parameter that distinguishes different thermal states if thermal equilibrium between thermodynamic systems is an equivalence relation.
Equivalence relation>>>> reflexive, symmetric and transitive
The Attempt at a Solution
The parameter should be temperature. First I could say that thermal equilibrium is reflexive (every thermal equilibrium of a system is in equilibrium with itself), and symmetric (if A is in equilibrium with B, then B is in equilibrium with A). If thermal equilibrium is an equivalence relation, then it must also be transitive (you can compare thermal equilibrium without thermal contact, aka 0th law of Thermodynamics). So you need a parameter to compare, and that parameter must be the temperature. Is this all I must answer?
PS> Sorry for my English