Is image and range of a LT the same thing ? (Why are they named like that)

[sid9221]

Is "image" and "range" of a LT the same thing ? (Why are they named like that)

If you were to tell me that "range" and "span" meant the same thing I would understand why, in english their meaning's describe a similar sort of idea but I can't see any similarity in range and image.

Why are they named that way ?

 

[Fredrik]

Suppose that f:X→Y is a function. If E is a set, then f(E) (i.e. the set of all f(x) such that x is in E) is called the image of E under f. Why "image"? Mainly because you have to call it something, and what else would you call it? It makes some sense to call it "image", if you think of X as a shape that's turned into something else by f.

The range of f is the specific set f(X), i.e. the image of the domain. So image and range don't mean exactly the same thing. The term "range" is natural, since f(X) is the set of all members of the codomain Y that are "reached" by the function f. f(X)={f(x)|x in X}={y in Y|there's an x in X such that f(x)=y}.

Also, span and range don't mean the same thing, but you probably know that. The span of a subset of a vector space is the smallest vector subspace that contains the set, or equivalently, the set of all linear combinations of members of the set.

 

[alexfloo]

The term 'image' as in 'image of a function' originated in a slightly different usage. The 'image of a point' under a function is the value of the function there, so if y=f(x), then y is the image of x. (We also say x is a 'preimage' of y.)

Another term that's often used is the 'image of a set under a function,' which refers to the set of all images of points in the set. For instance, the image of the iterval [-∞,-2] under the squaring operation is the interval [4,∞].

The 'image of a function' X→Y just refers to the image of X under the function. Range and image mean the same thing, and range has a more intuitive origin, like you mentioned. Usage of 'span' is pretty much isolated to linear algebra.

 

[jgens]

Suppose that f:X→Y is a function. If E is a set, then f(E) (i.e. the set of all f(x) such that x is in E) is called the image of E under f. Why "image"? Mainly because you have to call it something, and what else would you call it? It makes some sense to call it "image", if you think of X as a shape that's turned into something else by f.

I agree with all of this. I would add, however, that if you have a function f:X \rightarrow Y, then the image of f (denoted \mathrm{im} \; f) is the set f(X).

The range of f is the specific set f(X), i.e. the image of the domain. So image and range don't mean exactly the same thing. The term "range" is natural, since f(X) is the set of all members of the codomain Y that are "reached" by the function f. f(X)={f(x)|x in X}={y in Y|there's an x in X such that f(x)=y}.

I disagree with this. There are two common usages of the word range. One usage of the word range is the same as the image of a function, which is the usage you describe above. The other usage is for the codomain of the function. Both conventions are fairly common.

 

[SteveL27]

I disagree with this. There are two common usages of the word range. One usage of the word range is the same as the image of a function, which is the usage you describe above. The other usage is for the codomain of the function. Both conventions are fairly common.

Glad you mentioned that. It's very confusing to people. The image is a subset of the range, possibly a proper subset. So for example if f(x) = x², then we can say that the range of f is \mathbb{R}, and that would be an accurate statement. But not all of \mathbb{R} gets hit by f; so I would say that the image of f is the nonnegative reals. That's because I use the word image as in the image of a set; in other words the exact set of elements in the range that get hit by f.

And the term "codomain" was not in use when I was in school; and I actually have no idea what it means: image or range?

The moral is that writers should always be explicit when they use any of the terms range, image, and codomain; because not everyone agrees on their meaning.

 

[alexfloo]

The codomain of a function X→Y is Y, which I believe is the range in your usage. In my experience, however, range is used very inconsistently to refer to both meanings, but actually is more often synonymous with image.

 

[Fredrik]

I'm very surprised by these comments. Are there really books that define "range" so that f(X) can be a proper subset of the range of f? If you all have seen it, then I guess I have to believe it, but honestly, if it had been only two of you saying it instead of three, I would have assumed that you're both really drunk.

OK, I need to think this through to see if I can make sense of it. I will write down my thoughts. (Sorry, this will be a long post, but it will at least be useful to everyone who wants to understand the definitions of the term "function").

The idea behind the definition of "function" is that we should be able to think of a function f:X→Y as a rule that takes each member of the set X to exactly one member of the set Y. But to properly define the term in the framework of ZFC set theory, we must specify which set f is. Since there are many sets that we can think of as representing a rule like that, there are also many ways to make the concept of "function" (that we already understand on an intuitive level) mathematically precise. These are two similar, but not equivalent, ways to make the concept mathematically precise:

Definition 1

A set $f\subset X\times Y$ is said to be a function from X into Y, if
(a) For all $x\in X$, there's a $y\in Y$ such that $(x,y)\in f$.
(b) For all $x,x' \in X$ and all $y\in Y$, if $(x,y)\in f$ and $(x',y)\in f$, then $x=x'$.
X is said to be the domain of f. Y is said to be a codomain of f. f is also called the graph of f. So the function and its graph is the same thing.

Definition 2

A triple $f=(X,Y,G)$ is said to be a function from X into Y, if
(a) For all $x\in X$, there's a $y\in Y$ such that $(x,y)\in G$.
(b) For all $x,x' \in X$ and all $y\in Y$, if $(x,y)\in G$ and $(x',y)\in G$, then $x=x'$.
X is said to be the domain of f. Y is said to be the codomain of f. G is said to be the graph of f.

If we use definition 1, then a function has many codomains. If f is a function from X into Y and Y is a subset of Z, then Z is also a codomain of f. If we use definition 2, each function has exactly one codomain.

Definition 1 is slightly simpler, and seems to be the more popular one in the literature. It also seems desirable to have a definition that only admits one function that (for example) "takes every real number to its square". If we use definition 2, there are infinitely many such functions. For example, there's one with codomain {x in ℝ|x≥0}, one with codomain ℝ, one with codomain ℂ, etc.

But definition 2 at least has the advantage that it makes phrases like "f is surjective" unambiguous. If f is a function from X into Y, then it can only mean that f is surjective onto Y. But if we use definition 1, and Y is a subset of Z, it can also mean that f is surjective onto Z.

If we use one of these definitions, then it seems very strange to me to use the word "range" for the set Y. For example, the complex number -1+i would be in the range of the function $f:\mathbb N\to\mathbb C$ defined by f(n)=n+1 for all natural numbers n. This makes me wonder if there is any English word that would be less appropriate here than "range". Maybe those 7 words that George Carlin liked to talk about.

There is however a definition that doesn't require us to mention the codomain (or even the domain) at all. The trick is to first define an ordered pair (a,b) (where a and b are arbitrary sets) by (a,b)={{a},{a,b}}.

Definition 3

A set f whose members are ordered pairs is said to be a function if
(a) For all sets x,x',y, if $(x,y)\in f$ and $(x',y)\in f$, then $x=x'$.

If we use this definition, then it wouldn't make much sense to define the term "codomain" at all. We would define the domain X by
$$X=\Big\{x\in\bigcup\Big(\bigcup f\Big)\Big|\exists y\in\bigcup\Big(\bigcup f\Big)\ (x,y)\in f\Big\}.$$ The range f(X) can be defined similarly. I think this approach is much harder to understand (we need to know things like why ⋃(⋃f) is a set), so I wouldn't recommend it. The concept of "codomain" seems to be unnecessary, but if it makes the definitions simpler, then I don't mind using it.

Conclusion: I can see why someone might want to leave out the word "codomain", but not why anyone would want to define "range" differently. To be honest, I'm thinking that if it's really common to call the codomain "the range", then it's probably like the pronunciation "nucular" (damn you Jack Bauer) or the word "irregardless". It started with a misunderstanding, and then people started copying the mistake. Finally, so many people were doing it wrong that we started saying that it's not wrong anymore.

By the way, someone who doesn't have "codomain" in their vocabulary, and defines the range of f as the set f(X), shouldn't have any use for the term "surjective" (="onto"), because every function is surjective onto its range.

One of the reasons why I have no idea what terminology is common in English is that I first learned this from a book written in Swedish. The sets X,Y and f(X) were respectively called "definitionsmängd" (=definition set), "målmängd" (target set) and "värdemängd" (value set). I think those terms are much more intuitive than the English ones.

 

[alexfloo]

I believe definitions 1 and 3 are equivalent: given a function according to definition 3, we simply let X = {x:there exists some y s.t. (x,y) in f}. Then we have sufficient information to apply definition 1, and see that they agree. (The reverse agreement is trivial.) X never needs to be specified, because that information is encoded in the graph (the set f, that is). It's only Y that is sometimes necessary to make explicit, if we want to talk about surjectivity.

Really the only difference between 1/3 and 2 is that 1/3 make the codomain explicit. The 2-functions can be considered as functions from f into our universe of consideration, which are necessarily never surjective. However, the codomain is key anytime we want to consider an algebra of functions (that is, in any category), since each object has a distinct identity map. (If distinct objects had the same identity endomorphism, we would probably find some odd pathologies.)

 

[alexfloo]

By the way, I tutor a couple of students a precalculus class at my university, and one of the topics covered is "domain and range of functions" (always real-valued functions of a real variable). A function is not formally defined: it's considered to be an algebraic expression. The domain is either specified, or taken to be the "natural domain" of the expression which is the largest subset of R on which the expression is defined. Codomain is never specified, and the "range" sought is *always* the image.

However, this interpretation doesn't quite disagree with either definition of "range," since we don't explicitly have a codomain.

 

[jgens]

I'm very surprised by these comments. Are there really books that define "range" so that f(X) can be a proper subset of the range of f?

Yep. This is why range is an awful term. If you check the wiki page on the term "range" they mention the ambiguity in its usage there too.

Definition 1 is slightly simpler, and seems to be the more popular one in the literature.

Depends on which literature you are talking about. With definition 1, the functions f:\mathbb{R} \rightarrow \mathbb{R} and g:\mathbb{R} \rightarrow \mathbb{R} \setminus \{0\} given by f,g:x \mapsto e^x are equal; however, almost all mathematicians would say that these functions are not equal. So the definition most commonly used is certainly definition 2.

 

[alexfloo]

Depends on which literature you are talking about.

I've seen definition 1 used many times in contexts where, later, it becomes clear that definition 2 was meant. I think authors often feel its sufficient to suppress X and Y in the ordered triple, and consider them to be implicit in the notation "X→Y."

 

[Fredrik]

Depends on which literature you are talking about.
...
So the definition most commonly used is certainly definition 2.

That's certainly possible. My comment was only based on my own experience. I think that every time I've seen a definition written out, except at Wikipedia, it has been definition 1. I think I've seen it in a couple of books on set theory, and at least one on functional analysis, but I didn't even bother to check if I remember that right.

On the other hand, phrases like "Consider a surjection $\pi:E\to B$" don't even make sense if we use definition 1, and such phrases are certainly common in the literature. So it almost seems like textbook authors use definition 1 whenever they feel like it, and definition 2 whenever they feel like it. I wouldn't be surprised if we could find examples of two sentences from the same book, such that one only makes sense if we use definition 1, and the other only makes sense if we use definition 2.

 

[Fredrik]

A function is not formally defined: it's considered to be an algebraic expression.

I have noticed that a lot of people who ask questions here still think about functions this way. I have many times thought that this is the main reason why they can't understand the thing they're asking about, or my first reply to them. Hm, it's always hard to think of a good example. I guess it's when I say things like this:

If you understand ordinary derivatives, partial derivatives are really nothing new. $D_2f(x,y)$, i.e. the value at (x,y) of the partial derivative of f with respect to the second variable, is just the value at y of the ordinary derivative of the function $y\mapsto f(x,y)$.​

Some people aren't familiar with the "mapsto" arrow either, so that's another complication, but even if I say "the function that takes y to f(x,y)" instead of $y\mapsto f(x,y)$, they still don't understand what I'm saying.

This is an example of something they will usually understand when I explain it, but it's amazing how many times I've had to explain it:

f denotes a function. f(x) denotes a number in the range of that function. What number that is depends on the value of x, but that doesn't mean that f(x) is a function. It's just a number. f' is another function, called the derivative of f. f'(x) denotes the value of f' at x.​

 

[espen180]

I would argue that if f\,:\, X\rightarrow Y and Y\subseteq Z, then in order to get a "corresponding" function from X to Z we would need to compose f with the inclusion i\,:\, Y\rightarrow Z given by i=\mathrm{id}_Z|_Y, the result being a new function f^\prime=i\circ f \neq f. In fact the preceeding nonequality should be meaningless since the domains and codomains don't match. In other words, that the definition of a function includes the codomain, but it otherwise given as in Definition 1 above. Thus the term "surjective" is unambiguous.

 

[Fredrik]

I would argue that if f\,:\, X\rightarrow Y and Y\subseteq Z, then in order to get a "corresponding" function from X to Z we would need to compose f with the inclusion i\,:\, Y\rightarrow Z given by i=\mathrm{id}_Z|_Y, the result being a new function f^\prime=i\circ f \neq f. In fact the preceeding nonequality should be meaningless since the domains and codomains don't match. In other words, that the definition of a function includes the codomain, but it otherwise given as in Definition 1 above. Thus the term "surjective" is unambiguous.

What you're saying about the inclusion is automatic if we use definition 2. You seem to prefer a version of 1 that's modified to be practically equivalent to 2. So how would you modify it? By saying essentially the same things I said in 1, but reserving the term "function" for the pair (f,Y) instead of the set f (which is a subset of X×Y and therefore also a subset of X×Z)? Then we still lose the ability to write things like f⊂g, which is one of the advantages of definition 1.

What you're suggesting can also be described as a version of 2, in which we're talking about (G,Y) or (Y,G) instead of (X,Y,G). Such a definition would make sense, since X (the domain) can be recovered from G (the graph). But I don't think I would prefer this over my definition 2. That's of course just a matter of taste.

By the way, I don't think it's obvious that we want the codomain to be part of the function. I wrote a post half an hour ago where I found myself saying "Now consider the function f defined by f(x)=(sin x)/x for all x>0". I find it a little annoying that I would actually have to either change "the" to "a" or specify the codomain for the sentence to make sense.

There is something annoying about both kinds of definitions (definitions that include the codomain and definitions that don't), but I don't think it ever matters for anything other than the precise choice of words we need to use in simple sentences like that.

 

[espen180]

What is the significance of being able to write f⊂g? It seems to denote a restriction, which in any case can be accomplished by performing an inclusion from a subset of X and composing with f.

I think that the term "function" may be best suited for the pair "(f,Y)", as you suggested. I think that terms like "surjective" should be a property of the function, and not depend on an arbitrary choice.

In our example, I don't think you would need to specify the codomain for the function to make sense, only if you want to discuss properties depending on the codomain, like surjectivity. Otherwise I think the reader would (naturally?) assume that it is R.

Another example, statements about f having extremal points may depend on the codomain being compact.

 

[jgens]

There is something annoying about both kinds of definitions (definitions that include the codomain and definitions that don't), but I don't think it ever matters for anything other than the precise choice of words we need to use in simple sentences like that.

In some branches of mathematics, like category theory, which definition you use for functions is extremely important; this is especially true if you use an arrows only approach to categories and want to recover many of the usual concrete categories we talk about.

 

[Fredrik]

What is the significance of being able to write f⊂g? It seems to denote a restriction, which in any case can be accomplished by performing an inclusion from a subset of X and composing with f.

I don't think it is very significant. A function g is said to be an extension of f if dom f⊂dom g and f(x)=g(x) for all x in dom f. If we use definition 1, then g is an exension of f precisely when f⊂g, and I suppose things like that could be considered "nice".

I think that terms like "surjective" should be a property of the function, and not depend on an arbitrary choice.

This seems to be the best reason to prefer a definition that includes the codomain in the function. On the other hand, if we use a definition that doesn't, we don't seem to need the term "surjective" at all. The set theory book by Hrbacek & Jech uses this approach. I have a copy on my computer, and I can't even find the word "surjective" in it. Their definition of "function" is essentially the same as my definition 3, and they define "injective" right after "function".

In our example, I don't think you would need to specify the codomain for the function to make sense, only if you want to discuss properties depending on the codomain, like surjectivity. Otherwise I think the reader would (naturally?) assume that it is R.

Yes, they would almost certainly assume that it's ℝ. I can be a bit of a language nazi though, so I don't like to say things that strictly speaking don't make sense, even when the intended meaning is obvious.

 

[Rising Eagle]

I think we should go back to the definition of surjection and its relationship to the mapping between two sets. The two sets must have explicitly defined membership before an author's meaning is clear. Given a set X with members x and a set Y with members y, a surjection is a mapping where ALL the members of X are mapped to ALL the members of Y. It is allowable that the same y in Y can be mapped from multiple x in X as long as ALL the members in the set Y have at least one member from X partnered with it. In any mapping, by definition, ALL the members of X are mapped. We don't need to worry about the distinction between functions and relations here, only the generic idea of mapping between sets. Now we can define domain, preimage, range, image, and codomain:

Domain is ALL the members of X and X is the exact set of members to be mapped
NOTE: sometimes X is a subset of a larger set (e.g., a finite interval on reals), but the larger set is NOT the domain and is never referred to in a mapping definition. A mapping definition always refers to X and only X, the exact membership to be mapped.
Preimage is the exact subset of members of X specified by an inverse mapping of a subset of members in Y. It is not a synonym for domain, however the preimage is often equal to the domain.
NOTE: if the inverse mapping is from the image or a superset of the image, then the preimage is equal to the domain. If the inverse mapping is applied to a subset of the image, the preimage may exclude some members in the domain. Preimage is not an exact dual of image as the image is always a mapping from the entire domain.
Range is the exact subset of members in Y that have at least one member from domain X partnered with it
NOTE: if the exact subset is not the entirety of Y, then the mapping is not surjective
Image is an exact synonym for range, the terms are interchangeable
Codomain is the entirety of Y which may or may not be fully included in a given mapping. It is always a superset of the range and wil be equal to the range in the cases where the mapping is a surjection

For completeness, I will define injection, surjection, onto, one-to-one, bijection, function, relation, monotonic

Injection is a mapping where each member of Y has one or less member from X as a partner
Surjection is a mapping where the entire membership of Y has one or more members in X as a partner
Onto is an exact synonym for surjection, the terms are interchangeable
One-to-one is a mapping where each member of X has exactly one member of Y as a partner and each member of Y has one or less member from X as a partner; the mapping is also injective
Bijection is a mapping that is both one-to-one and onto; the mapping is also injective and surjective
Function is a mapping where each member of X has exactly one member of Y as a partner, but each member of Y may have none, one, or more than one member in X as a partner
Relation is any valid mapping
Monotonic is a one-to-one mapping where set members are numerical and the graph contains only positive and zero slopes or negative and zero slopes, but not a combination of the two; this is a function and is sometimes one-to-one and sometimes onto and bijective.

 

[jgens]

Range is the exact subset of members in Y that have at least one member from X partnered with it
NOTE: if the exact subset is not the entirety of Y, then the mapping is not surjective
Image is an exact synonym for range, the terms are interchangeable

Not exactly true. We have been over this already. There are two common usages of the term range. One of them is the usage you mention above and another is that the range is the codomain of a function.

Function is a mapping where each member of X has exactly one member of Y as a partner, but each member of Y may have none, one, or more than one member in X as a partner

Since you are using 'mapping' as a primitive notion, would you care to define that for us?

Monotonic is a one-to-one mapping where set members are numerical and the graph contains only positive and zero slopes or negative and zero slopes, but not a combination of the two; this is a function and is one-to-one (in most cases it is also onto and bijective)

This is a terrible definition of monotonic. A function f:\mathbb{N} \rightarrow \mathbb{R} can be monotonic but there is no well-defined slope. A monotonic function is also not necessarily one-to-one; for example, every constant function \mathbb{R} \rightarrow \mathbb{R} is monotonic but not one-to-one. Lastly, I am fairly certain that almost all monotonic functions are not bijections.

 

[Fredrik]

Preimage is an exact synonym for domain, the terms are interchangeable

I wouldn't define preimage=domain. I would define the term by saying that the preimage of a set E⊂Y is the set f^-1(E). So the domain is equal to the preimage of any set that contains the range. (I define the range as the set f(X)).

One-to-one is a mapping where each member of X has exactly one member of Y as a partner and each member of Y has one or less member from X as a partner; the mapping is also injective

I think it's common to define "one-to-one function"="injective function" and "one-to-one correspondence"="bijective function".

 

[Rising Eagle]

Not exactly true. We have been over this already. There are two common usages of the term range. One of them is the usage you mention above and another is that the range is the codomain of a function.

It is my humble opinion that Codomain became common more recently then Range and was introduced as an additional term specifically to refer to the entire set of Y and not the Range (a subset of Y being mapped to). If Range is used as a synonym for Codomain, it is redundant and leaves us to need yet another term to refer to the members of Y being mapped to. When it is used to mean Codomain, it is by mistake. It is my belief that the original use of Range was used as a dual to the term Domain which refers to the members of X being mapped. It just happens that all the members of X are always mapped as a matter of convention, so it looks at first glance as if the duality is broken, but it really is not. If we want to refer to a subset of X being mapped with some members left unmapped, we should use the term preRange or some such as a dual to Range. We have that with preimage and image. As it is now, both Range and Codomain are duals to Domain, each for a different property. Range is dual because the Domain is all members of X to be mapped from and Codomain is dual because Domain is the entire set X. As conventionally used, a mapping is always defined with a "preRange" being equal to the Domain, i.e., all members of X are mapped whenever a mapping is being defined. So I don't believe there are two uses of the term Range, just a second misuse or confused usage of the term.

Since you are using 'mapping' as a primitive notion, would you care to define that for us?

Sorry to leave out a definition for Mapping:

Mapping is a correspondence between two sets where a partnership is specified between a given member of one set and a given member of a second set. The list of partnerships is the mapping.

 

[Rising Eagle]

This is a terrible definition of monotonic. A function f:\mathbb{N} \rightarrow \mathbb{R} can be monotonic but there is no well-defined slope. A monotonic function is also not necessarily one-to-one; for example, every constant function \mathbb{R} \rightarrow \mathbb{R} is monotonic but not one-to-one. Lastly, I am fairly certain that almost all monotonic functions are not bijections.

No doubt that it is horrible. I have chosen to use casual language in this case because it is more intuitive and reminds us of the common uses and examples of the term in earlier introductions to math. I threw it into give physical imagery to my other definitions. Of course you are correct. I changed the last part for you so that they will be more correct.

 

[jgens]

Whether or not you think the term range as a synonym for codomain is a mistake, it does not change the fact that it is another usage for the term. There are well-respected mathematicians who range and codomain interchangeably, so it is naive to say that there is just one proper usage.

In any case, your definition of mapping too vague to be of much use. It sounds like you are trying to define the graph of a function or something like that, but are avoiding use of any of the language of set theory that could clarify what you mean.

 

[Rising Eagle]

I wouldn't define preimage=domain. I would define the term by saying that the preimage of a set E⊂Y is the set f^-1(E). So the domain is equal to the preimage of any set that contains the range. (I define the range as the set f(X)).

Of course you are correct. If the inverse mapping is from the image or a superset of the image, then the preimage is equal to the domain. If the inverse mapping is applied to a subset of the image, the preimage may exclude some members in the domain. Preimage is not an exact dual of image as the image is always a mapping from the entire domain. I will add a note to that effect.

I think it's common to define "one-to-one function"="injective function" and "one-to-one correspondence"="bijective function".

Here again you are correct. One-to-one function is one and the same with an injection. I guess I wanted to spell out a "function mapping" presumption I made implicitly in my def for injection by explaining that in X, each member is mapped only once. In a relation mapping, each member of X may be mapped more than once.

 

[alexfloo]

It is my humble opinion that Codomain became common more recently then Range and was introduced as an additional term specifically to refer to the entire set of Y and not the Range (a subset of Y being mapped to).

In category theory, a relatively new field, the prefix co- commonly signals the corresponding concept in reverse. Indeed, the codomain in the category Set is exactly the domain of the corresponding morphism in the dual category Set*.

This strongly suggests that the term codomain originated in the last 50 years, as you indicated.

 

[Rising Eagle]

Whether or not you think the term range as a synonym for codomain is a mistake, it does not change the fact that it is another usage for the term. There are well-respected mathematicians who range and codomain interchangeably, so it is naive to say that there is just one proper usage.

Couldn't disagree more. The use of terms must either be agreed upon or we all must define them every single time a discourse is undertaken. Right is right. Wrong is wrong. Communication is very inefficient if we don't iron out incorrect usage and meanings of agreed upon terms. I understand different usages in Physics and Math, but within the math community, we should be in agreement or find a good reason to permit an unconventional use of a basic and widespread term. I don't believe we have a good reason here. This whole thread is based on a confusion caused by loose usage of the term that started some time ago. It should have been corrected then. Since it is out there, still causing confusion, we better get started fixing it here and now. Maybe we need a watch list with warnings of deviations from conventional use, just to keep such confusion to a minimum.

It was also pointed out above in this thread that the same publication/same author would use one sentence implying one definition and another implying another definition of the same term. That would be an example of a mistake on the author/s part, not an acknowledgment that both uses are correct.

In any case, your definition of mapping too vague to be of much use. It sounds like you are trying to define the graph of a function or something like that, but are avoiding use of any of the language of set theory that could clarify what you mean.

Yes I have avoided set notation. I prefer intuitive notions properly stated (though yes I make mistakes) because getting bogged down in notation can distract from a full grasp of the idea. My informal definition of mapping should be quite adequate for most to grasp the definitions of other terms I presented earlier.

A graph is indeed a mapping, however, I was focused on specifying generic members of general sets in my definition of mapping, and not on sets whose members are numerical.

 

[jgens]

Couldn't disagree more. The use of terms must either be agreed upon or we all must define them every single time a discourse is undertaken. Right is right. Wrong is wrong. Communication is very inefficient if we don't iron out incorrect usage and meanings of agreed upon terms. I understand different usages in Physics and Math, but within the math community, we should be in agreement or find a good reason to permit an unconventional use of a basic and widespread term.

I agree that terms all need to be well-defined, but there are a plethora of terms that different mathematicians will use to mean different things. Take the term manifold for example. Some people will require a manifold to be second-countable and Hausdorff, while others will relax one or both of these conditions. And the fact is that there is no standard set in stone definition of which usage is correct. The reader will just have to pay attention to which convention the author is using. This is the same as the usage of the term range.

Yes I have avoided set notation. I prefer intuitive notions properly stated (though yes I make mistakes) because getting bogged down in notation can distract from a full grasp of the idea. My informal definition of mapping should be quite adequate for most to grasp the definitions of other terms I presented earlier.

Also as a note, I think everyone commenting in this thread has a good intuitive graps of functions, so there is no need to eschew the simplifications afforded by the use of set theory.

 

[espen180]

This seems to be the best reason to prefer a definition that includes the codomain in the function. On the other hand, if we use a definition that doesn't, we don't seem to need the term "surjective" at all. The set theory book by Hrbacek & Jech uses this approach. I have a copy on my computer, and I can't even find the word "surjective" in it. Their definition of "function" is essentially the same as my definition 3, and they define "injective" right after "function".

This may be unrelated, but wouldn't the related term "epimorphism" also be sacrificed? How would we prove theorems like "Any ring homomorphism from Z to a ring R can be extended uniquely to a homomorphism from Q to R" or similar theorems if we can't use concepts like codomain and epimorphism?

Also there is a theorem that any function between sets can be factored into a surjection followed by a bijection followed by an injection. What would happen to this theorem?

 

[alexfloo]

Epsen, epimorphism is a term primarily used within the context of a category where the meaning can be expressed otherwise.

For instance we might alternatively choose to say that a map is "an epimorphism to Y" but not an epimorphism to some larger set Y'. This would support the interpretation of a function as a relation with certain properties. Furthermore, for every function, there would be a unique set to which it is an epimorphism: it's image.