functions

[markosheehan]

is the function x²+3 surjective for real numbers. how do you test for surjectivity in general?

[ajb]

You should think about solutions to

 

[math] x^2 + 3 - y =0[/math]

 

Can this be solved for any real y? Or can you find at least one y where this equation cannot be solved?

[mathematic]

If x is real, [latex]x^2+3 \ge 3[/latex]

[ajb]

If x is real, [latex]x^2+3 \ge 3[/latex]

I was hoping markosheehan would reach such a conclusion by himself... I think this question could be homework.

[blue89]

is the function x²+3 surjective for real numbers. how do you test for surjectivity in general?

 

May I ask what that means ,"surjective" ??

[Country Boy]

 

May I ask what that means ,"surjective" ??

 

A function, f, from set A to set B, is said to be "surjective" (also called "from A onto B") if and only if, for every y in B there exist x in A such that f(x)= y.

 

A function, f, from set A to set B, is said to be "injective" (also called "one to one") if and only if, whenever f(x)= f(y), x= y.

 

A function, f, from set A to set B, is said to be "bijective" if and only if it is both surjective and injective (for every y in B, there exist one and only one x in A such that f(x)= y). A function is "invertible" if and only if it is bijective.

Edited August 27, 2016 by Country Boy

[studiot]

 

A function, f, from set A to set B, is said to be "surjective" (also called "from A onto B") if and only if, for every y in B there exist x in A such that f(x)= y.

 

A function, f, from set A to set B, is said to be "injective" (also called "one to one") if and only if, whenever f(x)= f(y), x= y.

 

A function, f, from set A to set B, is said to be "bijective" if and only if it is both surjective and injective (for every y in B, there exist one and only one x in A such that f(x)= y). A function is "invertible" if and only if it is bijective.

 

Nice and complete +1

[taeto]

I was drawn to the title of this somewhat dated thread.

Apparently to deduce the properties of a function you have to know not only "what it does", but also its exact domain and codomain; the sets on which the function is defined, and the set in which its values are found, respectively. You might think that the most efficient method of "knowing" everything about a function F is to have a complete table of the values of (x,y) for which y = F(x) holds. Even that does not give you complete information, since you wouldn't know how to answer questions about surjectivity and bijectivity of F without knowing the intended codomain, and that information isn't there.

So I have wondered what is the best way to explain a function, and what is the best way to define formally what it means for something to be a function?

I bought a copy of "The Handbook of Mathematics" and sought for the definition that it gives, but even though this is a pretty thick book, it didn't have any. I thought that was a little weird.

An analysis textbook explained that a function is given by a "formula". That was weird too, since it appears that mapping x to x+x or to 2x as a function from R to R should mean the same function, despite the difference between those two expressions. (The same textbook also occasionally would assume the choice axiom, which is used to produce functions that have no formulas at all.)

Is there any good answer to the question, what is a function really?

 

[studiot]

3 hours ago, taeto said:

That was weird too, since it appears that mapping x to x+x or to 2x as a function from R to R should mean the same function, despite the difference between those two expressions

This is a very good subject to consider as it highlights an important property of functions, that of being single valued.

Consider the function

[math]f(x):x = \sqrt 4 [/math]

Now the point about this very simple example is that 4 has two square roots, -2 and +2.

So if we allowed the function we call the square root function to have two values then we could justifiably say that for f(x) :  x + x = 0, but that f)x): 2x = +4 or -4.

 

We can develop this further if you like and also investigate other properties of functions.

[uncool]

In terms of set theory, a function is a set of ordered pairs (x, y) such that for each x, there is a some unique y such that (x, y) is in the set. 

[taeto]

2 minutes ago, uncool said:

In terms of set theory, a function is a set of ordered pairs (x, y) such that for each x, there is a some unique y such that (x, y) is in the set. 

      It is indeed. But then how do you answer the question about surjectiveness? You can assume that the range is known; the set of all y that appear as values. How do you identify the codomain?

28 minutes ago, studiot said:

This is a very good subject to consider as it highlights an important property of functions, that of being single valued.

Consider the function

f(x):x=4–√

Now the point about this very simple example is that 4 has two square roots, -2 and +2.

So if we allowed the function we call the square root function to have two values then we could justifiably say that for f(x) :  x + x = 0, but that f)x): 2x = +4 or -4.

 

We can develop this further if you like and also investigate other properties of functions.

       There should not be any worry about functions being single valued. If you use a square root notation, it is automatically assumed that you mean the principal root, which is nonnegative in the real case, in particular. That is not the point.Other than that, your example is better than mine, since if you take a function to mean a "formula", then assigning the value √4 to x still seems different from assigning the value 2 to x, since the "formulas" for the assigned values are different.

      You use the term "function". What do _you_ mean exactly by this notion? That was my query, and I think also in the spirit of the OP's original question. 

[studiot]

12 minutes ago, taeto said:

      It is indeed. But then how do you answer the question about surjectiveness? You can assume that the range is known; the set of all y that appear as values. How do you identify the codomain?

 

That was not your question, although it would have been addressed, but I note that you don't want a discussion on that subject.

Note also that the range is a different thing from the co-domain.

[taeto]

1 minute ago, studiot said:

 

That was not your question, although it would have been addressed, but I note that you don't want a discussion on that subject.

Note also that the range is a different thing from the co-domain.

     I don't know what you mean; this was exactly the first part of my question: how do you define "function" in a way so that you can identify both "what it does" and also whether it has a property like being surjective, which was in the question by the original OP. The fact that the range is not the codomain is obviously the important reason why you cannot do that from the graph of the function alone. 

What is it that you think that I do not want a discussion about? Sorry to seem dismissive about your thoughts on singled-valuedness, I just did not think it relevant.

 

 

[studiot]

I think we cross posted since When I last looked you had replied to uncool, apparantly but not bothered with my earlier post.

However I see that you have since replied.

 

So let's start (again) with some definitions.

 

There is no such thing in general as the co-domain.

A valid co-domain is any set that includes all the values of a function.
There is no requirement for it not to include any members that are not values of the function.

 

That honour is given to a more restricted set called the range which is the set of values taken on by a function as its argument varies through its domain.
Thus the range is the image of the domain in the co-domain.

Now if every member of the of the co-domain is the image of at least one member of the domain then the function is said to be surjective.
In this case the range and the co-domain are the same or coincident.

As an example the mapping from the set of all men to the set of married women is surjective (modern gender relations aside). But there remain men in the domain who are not married, that is their function value is null.

It's tricky to get all the definitions to match sweetly, which is why so much thought was put into it by many clever people.

How are we doing?

Edited June 13, 2018 by studiot

[taeto]

27 minutes ago, studiot said:

There is no such thing in general as the co-domain.

A valid co-domain is any set that includes all the values of a function.

That honour is given to a more restricted set called the range which is the set of values taken on by a function as its argument varies through its domain.
Thus the range is the image of the domain in the co-domain.

Now if every member of the of the co-domain is the image of at least one member of the domain then the function is said to be surjective.
In this case the range and the co-domain are the same or coincident.

As an example the mapping from the set of all men to the set of married women is surjective (modern gender relations aside). But there remain men in the domain who are not married, that is their function value is null.

It's tricky to get all the definitions to match sweetly, which is why so much thought was put into it by many clever people.

How are we doing?

Brilliantly, I think! Thanks for your input.

You have a different understanding from me of the properties of a function, judging from your example. Are you certain that you would argue that the function that maps x to 1/x has the entire set of real numbers as its domain, and it just happens to be the case that there is an element 0 in the domain that has null function value? That seems controversial, and not the usual understanding.

And you suggest that a co-domain is not (necessarily) fixed in the description of a function. In which case it is moot to ask about surjectivity, because every function is surjective if only you choose its co-domain small enough. Also bijectiveness would be equivalent to injectiveness, by the same token. I can agree that this is also what I get from the raw set-theoretic definition of a function stated by uncool. It still does not seem right; it should be possible to ask questions about surjective versus non-surjective. I just do not see how to best do it in the way of a simple explanation, or how to do it exactly formally either. 

[studiot]

34 minutes ago, taeto said:

Brilliantly, I think! Thanks for your input.

You have a different understanding from me of the properties of a function, judging from your example. Are you certain that you would argue that the function that maps x to 1/x has the entire set of real numbers as its domain, and it just happens to be the case that there is an element 0 in the domain that has null function value? That seems controversial, and not the usual understanding.

And you suggest that a co-domain is not (necessarily) fixed in the description of a function. In which case it is moot to ask about surjectivity, because every function is surjective if only you choose its co-domain small enough. Also bijectiveness would be equivalent to injectiveness, by the same token. I can agree that this is also what I get from the raw set-theoretic definition of a function stated by uncool. It still does not seem right; it should be possible to ask questions about surjective versus non-surjective. I just do not see how to best do it in the way of a simple explanation, or how to do it exactly formally either. 

A long time ago I was taught that the full specification of a function comprises three things.

Specification of the Domain

Specification of the Co-domain

Definition of the rule of mapping.

So it is up to the specifier of the function to specify these and by choosing a different domain / co-domain / rule we can vary the function or choose a different one.

You mentioned a hyperbola, which is officially a conic with two branches,(unlike say a parabola that only has one branch) but by restricting the domain we can make it a 'new' function that has only one branch.

 

Would you like to compare the terms injective, surjective and bijective?

 

Note uncool talks of ordered pairs which implies there are only two variables, but of course you can extend this count.

 

Edited June 13, 2018 by studiot

[taeto]

6 minutes ago, studiot said:

Specification of the Domain

Specification of the Co-domain

Definition of the rule of mapping.

That seems a standard way of saying it alright. But it is not what I mean. 

I would almost prefer the way in which uncool puts it, because now in your suggestion you get into a problem with this "rule" notion. I can concede that if by "rule" you simply mean: the rule is that y = f(x) holds if (x,y) is the point on the graph of f that has x as its first component. Then that might seem fine as a rule, except it appears quite roundabout, besides being cyclic. But be aware that this "rule" thing can never be interpreted as saying that you have a "formula" for computing y once you are given x as input. Once you apply the Axiom of Choice you will be 100% sure to get heaps of functions none of which can be computed by any formula.   

Now I try a definition that makes sense to me.

A function f is a pair f = (F,B), for which there is a set A such that F is a set of pairs (x,y) with x in A and y in B, such that for every x in A there is precisely one pair in F with x as its first component. If (x,y) is this pair then we write y = f(x). The set A is called domain of f, the set B is called co-domain of f, and F is called the graph of f. The range f(A) of f is the set of y in B for which there exists x in A with (x,y) in F.  We say that f is injective if (x1,y) and (x2,y) both in F implies x1 = x2, and that f is surjective if f(A) = B, and that f is bijective, if f is both injective and surjective.

I think that would be alright in set theory, unless I missed something. But it is horribly cumbersome. I can imagine that you agree at least on this point. Have you seen it done better, while roughly capturing the same meaning?

 

[studiot]

I thought I had pointed out that pairs may not be adequate.

Nor need a function be analytic or specified by an equation.

All that is required is that every time you apply it to a member of the domain set you get the same answer.
So it can't be a table of experimental results directly. But it could be a specific table such as a logic truth table (Cayley table)
Or it could be a filter for instance one which has all the integers as the domain but selects only odd integers.

[taeto]

12 hours ago, studiot said:

I thought I had pointed out that pairs may not be adequate.

Nor need a function be analytic or specified by an equation.

All that is required is that every time you apply it to a member of the domain set you get the same answer.
So it can't be a table of experimental results directly. But it could be a specific table such as a logic truth table (Cayley table)
Or it could be a filter for instance one which has all the integers as the domain but selects only odd integers.

If you are thinking about functions of more than variable, then it seems fine to consider, say, a function of three real variables as a function of a single variable but defined on R^3. More generally a function of several variables can be viewed as a function of a single variable with a domain consisting of the suitable product of sets. In this sense the representation using just pairs (x,y) is quite adequate.

Otherwise I agree with what you say.

[studiot]

I'm glad we have lots of points of agreement and, in fact, I agree that technically your point about R3

[aside] Did you know this site has superscript and subscript, which are very useful? Looka long the top toolbar in the editor for X₂ and X².[/aside]

If you regard a function as a map from a set of all ntuples for some n (eg 3 in your case) to R

This is, of course, also the definition of a functional.

But I think this is somewhat disingenuous since you have to know all n numbers somehow.

Further complication comes when the Xs are not numbers so the map cannot be to R. For a simple example a function connecting the strain tensor to the stress tensor. The objects in both the domain and co-domain are ntuples or arrays.

 

It is really nice to chat amicably like this but it would also be nice to know where it is all leading?

It certainly goes to show how slippery some definitions are.

[taeto]

Thanks for the formatting tip; I have been too much in time constraints to notice the features.

I am not sure about disingenuity. I am happy as long as the domain is precisely defined, in the shape of a set. Doesn't matter whether sets of numbers or not, those are just the basic examples. To consider other sets does not present any further formal complexity.

My conclusion so far is this. For the typical informal explanation of a function you would have something like: let f : A -> B be the function given by f(x) = (something or other), where something can be anything that starting from an element x of A determines a unique element of B. Even when it is obtained from an application of the Axiom of Choice (though in that case the exact wording of the explanation probably is different). Then there is no problem with identifying the well-known properties; injectiveness, surjectiveness, bijectiveness, and for heck's sake, throw in continuity, differentiability, smoothness etc. as well, supposing they apply.

But for the formal explanation of a function I am not so sure. Q: Is the function x²+3 surjective? A: I don't know, what is your arithmetic? Q: natural numbers. A: no it isn't, you don't get 5 into your range. Q: wait, I meant rationals. A: no, you still don't get 5. Q: wait, I actually meant real numbers. A: the domain is all real numbers? Q: yes. A: what is your co-domain, also the real numbers? Q: yes,of course. A: no, then you do not get 2, please choose a different co-domain. Q: no, instead I want to choose a different domain now: all algebraic complex numbers, how about that? A: and the co-domain is still all real numbers? Q: yes, of course, when did I say otherwise, don't be a jerk. A: no, sorry, that does not even give you a function... And so this exchange can go on forever. Much like conversations in this forum, if I may make that observation. I hope you get the point that so long as you do not have a clear concept in mind of the meaning of a term, like "surjective", and it has to be explained relative to a lot of other circumstances that are arbitrary to the parties, then you are not getting any dialogue established.

But apparently I cannot pinpoint the formalization of "function" which avoids this. 

Edited June 14, 2018 by taeto
change of misprint