How much commutativity in associativity? Please correct me if I am wrong. By the definition, binary operation "+" on set S is associative if and only if, for all elements x, y, and z from S, the following holds: x + (y + z) = (x + y) + z. In other words, the order of operation is immaterial if the operation appears more than once in an expression. Now, operation "+" may be either commutative or not. Let us consider the later case. If "+" is not commutative, we have the following: x + (y + z) may be different from x + (z + y) and x + (y + z) may be different from (y + z) + x and x + (y + z) may be different from (z + y) + x. Thus, the result of x + (y + z), depending on the way the operation is performed may by different from the result of (x + y) + z. The whole problem disappears whenever "+" is commutative. So, can we really claim that "+" is associative without referring, maybe even not explicitly, to commutativity? Kocur.