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I have recently read a book of analysis, which starts with some stuff about Set Theory before moving on to functions. By reading the book, I realised (it was written in it) that the simple operations that we do (addition,multiplication) are actually functions.

Let's speak for the real numbers only:according to the book addition and multiplication are two functions from RxR to R, and actually functions are sets . So, addition of a pair (x,y) with x,y real numbers is a number that is the value of the function "addition" at the "point" (x,y) and is written x+y . So x+y=S((x,y)) if we name the function of addition S. The same applies to multiplication , so the value of the function M(=multiplication) is written xy and xy=M((x,y)).

So here comes the question: When we write x+y (example 5+3 etc) or xy , x+y or xy is actually one number and not an expression? To make myself more clear, x+y is not an expression that when evaluated gives the sum of x and y but a number that we, in order to find it have to add x and y? (I don't know if anyone else understands what i am asking!) So actually x+y ( or x+y+2z etc) is always one number, which is the "image" of a function(addition or multiplication) and we can write it in many ways because addition ( or multiplication) is not a "1-1" function? And we have just figured out some algorithms in order to find that image through calculations using the (x,y) pair , from which the image is "produced" , or generally transformate these pairs with others that have the same image , aslo making sure that the axioms referring to addition and multiplication are true?

To sum up the expression x+y (or a more complicated one) represents/is equal to one unique real number , so when we write x+y ,even if we don't calculate the sum, this still is equal to this unique number z with

z=S((x,y), and is not just an expression whose value is equal to z?

Are similar functions defined for operations in a vector space generally?

That's all(for now). Sorry for the weird language-English is not my mother-tongue. I hope that you have understood what i am trying to ask!(of course it may be just nonsense)

Thanks

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# Read a book of analysis

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