### Question about Associativity?

Posted:

**Sun Oct 30, 2011 6:07 pm UTC**{I screwed up and mistakenly called associativity, transitivity; I am talking about associativity. I HAVE CHANGED THE TITLE OF THE THREAD TO REFLECT THIS, but have left my errors intact in my post.}

I need a symbol for a "generic" operator; I don't know what the mathematical standard for that is, so for now I am going to use the "@" character as a symbol for a "generic" operator.

In my math classes, one has proved transitivity by proving that (a@b)@c=a@(b@c); this does prove transitivity over 3 operands, but can one prove that transitivity over 3 operands implies transitivity over any number of operands? To use 4 operands as an example, there are 5 or 6 possible cases to be proven:

((a@b)@c)@d

a@(b@(c@d))

(a@(b@c))@d

a@((b@c)@d)

The last case(s) is: (a@b)@(c@d); for all the operations that have been considered in my math classes, the operator is "memory-less"; ie, (a@b) has the same result whether it is done before or after (c@d) [likewise for (c@d)], which would make (a@b)@(c@d) a single case, thus resulting in 5 cases to be proven. However, it is possible to imagine that the operator is a function is a computer program, which means that the operation could have a memory, meaning that the result of (a@b) could depend whether it is done before or after (c@d) [likewise for (c@d)], which would make (a@b)@(c@d) 2 cases, thus resulting in 6 cases to be proven.

There are two things to consider. One is that I have only given the cases for 4 operands, but I am asking about the question of proving that transitivity over 3 operands implies transitivity over any number of operands.

The other is that it is possible that while it may be proving that transitivity over 3 operands implies transitivity over any number of operands for a "memory-less" operation, it may not be true that proving that transitivity over 3 operands implies transitivity over any number of operands for a operator with a memory. Considering that I have never been in a situation where I have needed to prove transitivity of an operation with a memory, I am far more interested in whether or not proving that transitivity over 3 operands implies transitivity over any number of operands for a "memory-less" operation, than I am for an operation with a memory.

I need a symbol for a "generic" operator; I don't know what the mathematical standard for that is, so for now I am going to use the "@" character as a symbol for a "generic" operator.

In my math classes, one has proved transitivity by proving that (a@b)@c=a@(b@c); this does prove transitivity over 3 operands, but can one prove that transitivity over 3 operands implies transitivity over any number of operands? To use 4 operands as an example, there are 5 or 6 possible cases to be proven:

((a@b)@c)@d

a@(b@(c@d))

(a@(b@c))@d

a@((b@c)@d)

The last case(s) is: (a@b)@(c@d); for all the operations that have been considered in my math classes, the operator is "memory-less"; ie, (a@b) has the same result whether it is done before or after (c@d) [likewise for (c@d)], which would make (a@b)@(c@d) a single case, thus resulting in 5 cases to be proven. However, it is possible to imagine that the operator is a function is a computer program, which means that the operation could have a memory, meaning that the result of (a@b) could depend whether it is done before or after (c@d) [likewise for (c@d)], which would make (a@b)@(c@d) 2 cases, thus resulting in 6 cases to be proven.

There are two things to consider. One is that I have only given the cases for 4 operands, but I am asking about the question of proving that transitivity over 3 operands implies transitivity over any number of operands.

The other is that it is possible that while it may be proving that transitivity over 3 operands implies transitivity over any number of operands for a "memory-less" operation, it may not be true that proving that transitivity over 3 operands implies transitivity over any number of operands for a operator with a memory. Considering that I have never been in a situation where I have needed to prove transitivity of an operation with a memory, I am far more interested in whether or not proving that transitivity over 3 operands implies transitivity over any number of operands for a "memory-less" operation, than I am for an operation with a memory.