- #1

Kocur

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**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.

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