Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example U = U (S, V,{ N j }), that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, U = U (S, V,{ N j }) for S, to get S = S (U, V,{ N j }).