zev:
So… these software applications I’m using were really written in “D”?
What exactly is the difference between an operator and a function? How is a uniary operator different from a function? I don’t see the difference. I see no difference between a binary operator on R and a function from R x R into R. For that matter, the differential operator is just a function whose domain is certain well-behaved functins and whose range is functions. Function, operator, transformation – they all look the same to me. What is the techinical distinction?
SCSimmons
You left out one property of natural numbers: The set of natural numbers is the smallest set that has the five properties that you listed. That is, if we have a set S that has the five properties you listed, the set of natural numbers is equivalent to a subset of S.
No.
The value of the expression C++ is the same as the value of the expression C. Maybe they should have called the language ++C.
And even that would be the character ‘D’ only if the value of the variable C happened to be the character ‘C’.
I suppose that C++ could evaluate to the string “D”, exactly when C is the string “D”.
So, in order to know what C++ is, you have to know what C is; and then they’re just the same thing except C++ carries a little extra baggage along with it. Perhaps the name of the language is apropos after all.
Confused people should go to the library and check out a book on Group Theory… Linear Algebra too while they’re at it. These will give you some good non-rigorous, rudimentary understandings of what on Earth operators are and how they’re different from functions. For example, in physics, any operator that is Hermetian is representative of an observable… but I digress.
The OP (assuming operations are what is meant instead of functions) doesn’t define the set they are concerned with well enough for me to answer the question. For example. the set could be all real numbers and all real cucumbers. You can define an operator that has the property that the operator operated on the number 2 gives the cucumber in my refrigerator. I’ll be damned if I know what kind of combination of +,* will give you that answer.
If we are to assume we’re dealing with a set of Real Numbers, then we can begin to behave a bit less humorously. If the question is, can we define finite piecewise operators to map the set of real numbers to itself (or a subset of itself) using simply addition and multiplication the answer is no (since the real numbers are not a group). However, for the set of natural numbers, the answer is yes (since the natural numbers are a group). Also, you may note that multiplication and addition don’t have to be related, but you have to include a concept of numerating your iterations in your reconstructed, piecewise operation if you want to define some relation (in other words, a three has to represent three members of the set as well as the element “3”). This is not a trivial problem, and I actually know of no solution (this is perhaps due to the fact that I have never taken a set-theory course). It really hasn’t bothered me that much, but it may come back to haunt me in the future.
Please, mathematicians… I’m a bit rusty in this area as it has been years since taking classes in this subject. Correct me if I’ve gone out into left-field.
Actually, I believe it’s the whole numbers (natural numbers and zero) that’s a group since zero is the additive identity. A thousand apologies.
Computer Scientists of my ilk generally consider zero as a natural number. Pretend that’s what you meant.
Operators, in the formal sense, are mappings between function spaces–see here. I don’t think that addition on the naturals is formally an operator, but hey, whatever.
2 * 3 doesn’t have to mean that, though. That’s just the recursive framework that we’re all familiar with. I’m not sure what multiplication would look like in an iterative setting, but I’m pretty sure it would look different.
The guy I had for computation theory didn’t. I didn’t know he was in the minority.
Although I’m not 100% certain how it affects your argument, the real numbers are a group under addition, at least according to the standard definition–we do agree on that, right? If not, what do you mean when you say “group”?
Neither the set of positive integers nor the set of non-negative integers forms a group under addition: elements do not have additive inverses in those sets.
Right, but to define * in terms of + in general requires recursion. You can define m3 as m+m+m, but then you haven’t defined , you’ve defined a unary postfix operator 3. You still don’t know what m4 is unless you define another unary postfix operator so that m4 = m+m+m+m. But using recursion, we can say m0 = m and m*(n+1) = m*n + m, thus defining * for all nonnegative integers.
Crap, not another mathworld link. every time someone links there I spend hours reading it, which means that I am backlogged on my SDMB hours, which means my work gets pushed aside. 
Yeah, I know that feeling. Damn site’s addictive.
Some people might claim that the only mathematical operation is counting and all the others are just algorithms to speed up the process.