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

  1. May 20, 2012 #1
    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 ?
     
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  3. May 20, 2012 #2

    Fredrik

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

    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.
     
    Last edited: May 20, 2012
  4. May 21, 2012 #3
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.
     
  5. May 21, 2012 #4

    jgens

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

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

    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.
     
  6. May 21, 2012 #5
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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 [itex]\mathbb{R}[/itex], and that would be an accurate statement. But not all of [itex]\mathbb{R}[/itex] 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.
     
  7. May 21, 2012 #6
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.
     
  8. May 21, 2012 #7

    Fredrik

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

    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. :smile:

    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.
     
  9. May 21, 2012 #8
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.)
     
  10. May 21, 2012 #9
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.
     
  11. May 21, 2012 #10

    jgens

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

    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.

    Depends on which literature you are talking about. With definition 1, the functions [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex] and [itex]g:\mathbb{R} \rightarrow \mathbb{R} \setminus \{0\}[/itex] given by [itex]f,g:x \mapsto e^x[/itex] are equal; however, almost all mathematicians would say that these functions are not equal. So the definition most commonly used is certainly definition 2.
     
  12. May 21, 2012 #11
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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."
     
  13. May 21, 2012 #12

    Fredrik

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

    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.
     
    Last edited: May 21, 2012
  14. May 21, 2012 #13

    Fredrik

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

    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.​
     
    Last edited: May 22, 2012
  15. May 22, 2012 #14
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    I would argue that if [itex]f\,:\, X\rightarrow Y[/itex] and [itex]Y\subseteq Z[/itex], then in order to get a "corresponding" function from X to Z we would need to compose f with the inclusion [itex]i\,:\, Y\rightarrow Z[/itex] given by [itex]i=\mathrm{id}_Z|_Y[/itex], the result being a new function [itex]f^\prime=i\circ f \neq f[/itex]. 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.
     
  16. May 22, 2012 #15

    Fredrik

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

    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.
     
  17. May 22, 2012 #16
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.
     
  18. May 22, 2012 #17

    jgens

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    Re: Is "image" and "range" of a LT the same thing ? (Why are they named 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.
     
  19. May 22, 2012 #18

    Fredrik

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

    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".

    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".

    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.
     
    Last edited: May 22, 2012
  20. May 22, 2012 #19
    Re: Is "image" and "range" of a LT the same thing ? (Why are they named like that)

    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.
     
    Last edited: May 22, 2012
  21. May 22, 2012 #20

    jgens

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

    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.

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

    This is a terrible definition of monotonic. A function [itex]f:\mathbb{N} \rightarrow \mathbb{R}[/itex] 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 [itex]\mathbb{R} \rightarrow \mathbb{R}[/itex] is monotonic but not one-to-one. Lastly, I am fairly certain that almost all monotonic functions are not bijections.
     
    Last edited: May 22, 2012
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