(adsbygoogle = window.adsbygoogle || []).push({}); How much commutativity in associativity?

Please correct me if I am wrong.

By the definition, binary operation "+" on setSis associative if and only if, for all elementsx,y, andzfromS, 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 ofx+ (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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How much commutativity in associativity

Loading...

Similar Threads for much commutativity associativity |
---|

B How does matrix non-commutivity relate to eigenvectors? |

I Splitting ring of polynomials - why is this result unfindable? |

I Is a commutative A-algebra algebraic over A associative? |

A Is the proof of these results correct? |

A Nonlinear operators vs linear |

**Physics Forums | Science Articles, Homework Help, Discussion**