What do you mean by saying that the bracketings are equivalent? That expressions with all possible bracketings are defined simultaneously and are equal when defined? It could be that a * (b * c) is defined while (a * b) * c is not; then the fact that * is associative says nothing for these a, b and c. Or are you saying that they are equivalent in the sense that only those expressions with all possible bracketings that are defined are equal?