Well! theoretically it must be right. This is the associative property of the addition. I hope from + you mean addition, and from () you mean priority of the operation.

As I used to be a theoretician in my former program (it is not that old), but I would like to bring your attention to application in this post.

Yes, + is the add operator and the parenthesis identifies the priority of the + operators. Anyway, I hope all readers of this blog knows the relation for division at least theoretically.

Any binary operation is theoretically defined on the Cartesian product of a set. Possessing or not possessing the associative property directly depends on (a) the set on the Cartesian product of which one defines the operation, and (b) the definition of the operation itself. One can easily define even the additive operation such that it is not associative (the additive operation is not necessary the regular addition defined on the set of real numbers!!!). So, the general answer is “no”. That is, the equality (a+b)+c=a+(b+c) does not necessary hold ( you can replace + by any binary operation as this is just a symbol). Finding a counter example is trivial.

Thanks dear Damoon for your insightful comment. Actually, I considered the simple addition operator that is commonly used in programming language and is taught in school, assuming it has the associative property. So I invite to check out my new post for more details.

You may want to assert it for division as well,

ReplyDelete\(a/(b/c)\)?\((a/b)/c\)

Well! theoretically it must be right. This is the associative property of the addition. I hope from + you mean addition, and from () you mean priority of the operation.

ReplyDeleteAs I used to be a theoretician in my former program (it is not that old), but I would like to bring your attention to application in this post.

ReplyDeleteYes, + is the add operator and the parenthesis identifies the priority of the + operators.

Anyway, I hope all readers of this blog knows the relation for division at least theoretically.

Counterexample of associativity of + operator!

ReplyDeletea=1

b=10^99

c=-10^99

I couldn't understand, what is the point?

DeleteI couldn't understand, what is the point?

DeleteCheck out the new post!

DeleteAny binary operation is theoretically defined on the Cartesian product of a set. Possessing or not possessing the associative property directly depends on (a) the set on the Cartesian product of which one defines the operation, and (b) the definition of the operation itself. One can easily define even the additive operation such that it is not associative (the additive operation is not necessary the regular addition defined on the set of real numbers!!!). So, the general answer is “no”. That is, the equality (a+b)+c=a+(b+c) does not necessary hold ( you can replace + by any binary operation as this is just a symbol). Finding a counter example is trivial.

ReplyDeleteThanks dear Damoon for your insightful comment. Actually, I considered the simple addition operator that is commonly used in programming language and is taught in school, assuming it has the associative property. So I invite to check out my new post for more details.

ReplyDelete