Defining differentitation and integration on functions

[Mr Davis 97]

I have a question concerning how how we define the differentiation and integration operators. Firstly, I know that functions are typically defined as an ordered triple triple ##(X, Y, f)## such that ##f⊆X×Y##, where ##x \in X## and ##f(x) \in Y##. This all seems nice and fine, but we also define the differentiation operator as ##\frac{d}{dx}: f \mapsto f'##. My confusion lies in the fact that ##\frac{d}{dx}## takes expressions such as ##5x^2 + 2 = f(x)## as inputs, not ##f##; e.g., ##\frac{d}{dx} (5x^2 + 1) = 10x## (Obviously, an ordered triple is not the input). The operator seems to take the image of ##x## under ##f##, which is ##f(x)##, not the function ##f## itself, which is the ordered triple. Why, then, do we define the differentiation operator as mapper of functions, when it takes ##f(x)##, an expression, as input, rather than ##f##, a function, as input? This question also applies to the indefinite integral operator, which supposedly maps ##f## to ##F##, where ##F' = f##.

 

[Stephen Tashi]

My confusion lies in the fact that ##\frac{d}{dx}## takes expressions such as ##5x^2 + 2 = f(x)## as inputs, not ##f##; e.g., ##\frac{d}{dx} (5x^2 + 1) = 10x## (Obviously, an ordered triple is not the input). .

You have to distinguish between what a human being takes as an input when he performs differentiation and the mathematical definition of a differential operator. Procedures for a human being to work problems in calculus are not mathematical definitions.

 

[GiuseppeR7]

Hi, i have basics knowledge of math but the defintion of a function is not an oredered triple but: given two sets D and C a function is a relation in DxC that have the property that if x is in D there is only one element in relation to that. So a function is a relation with a property not an ordered triple.
Of course the operator of differentiation is a function that take a function and gives as the output the famous derivative. the input can be 5x^2+1 since this is a function from R to R.

 

[Stephen Tashi]

but the defintion of a function is not an oredered triple but: given two sets D and C a function is a relation in DxC that have the property that if x is in D there is only one element in relation to that. So a function is a relation with a property not an ordered triple.

A function can be regarded as an ordered triple and may be defined as such. You have to mention sets D and C in the definition. So the function consists of the ordered triple (D,C,f) where D and C are sets and f is the relation. You can state most mathematical definitions without mentioning ordered lists of things, but the most formal way to state them is list the things involved and then state their properties.

 

[GiuseppeR7]

I can not argue with what you are saying since you know more than me. But i have to admit that this create very much confusion into my mind, for me a function IS a relation between two sets and thinking to a function as an ordered triple does not make sense to me since these two are different mathematical entities. Maybe you can describe a function as an ordered triple but not define with it. Can you suggest to me a good book about math?

 

[Stephen Tashi]

a function IS a relation between two sets and thinking to a function as an ordered triple does not make sense to me since these two are different mathematical entities. Maybe you can describe a function as an ordered triple but not define with it.

It's true that some texts define a function without saying it is an ordered triple. But other texts define it as such. You can define a function f as a relation on the cartesian product of two sets C and D without mentioning a list D,C,f of sets that are involved. You can also define the function as a list of 3 things (D,C,f) and state the properties of D,C,f. Since f is a set of ordered pairs and f appears within the odered list of 3 things (D,C,f) the meaning of a function as a relation is still conveyed.

Can you suggest to me a good book about math?

If you explain what mathematics you have already studied, perhaps someone can suggest a good text or online videos. I don't know about modern books because I've been out of school for 20 years.