confusion regarding the concept of a function

I was reading principles of mathematical analysis by Walter Rudin chapter 2 when a confusion about the definition of a function cropped up. (Read definition in comment below) I had thought that functions were maps from a set called the domain to another called the codomain. From what i knew before, these maps should not be one-many. Otherwise it would simply be a Relation not a function. But the book in its definition does not talk about the fact that functions should not be one-many. Infact it doesnt differentiate between a general mapping and a function and puts no restriction on what a function can be. So my question is where exactly is the mistake?

UPDATE: The discussion till now points that maybe the "an" word in the definition points towards uniqueness of image $f(x)$ of an element $x$ in the domain. I have also commented below regarding "well-defined functions" and "not well-defined functions". I think the notion of "well-defined"ness has got to do with differences in the output for the same input in various "forms" rather than just an element in the domain mapping to various elements in the range. Hence it does not in anyway interfere with the notion that a "function" cannot be one-many. Also to make things clear, Rudin uses the word "or" in conjunction with the word "mapping" and "function". Hence he is essentially saying that the mapping is a function and a function is a mapping. Which also implies that relations in general are not mapping. Please correct me if I've been wrong.

• As I don't have the book handy, I don't know the exact words Rudin uses. Knowing that might make it easier to answer the question. – Gerry Myerson Jan 18 '13 at 4:50
• Definition 2.1 Consider two sets $A$ and $B$, whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, an element of $B$, which we denote by $f(x)$. Then $f$ is said to be a function from $A$ to $B$ (or a mapping of $A$ into $B$). The set $A$ is called the domain of $f$ (we also say $f$ is defined on $A$), and the elements $f(x)$ are called the values of $f$. The set of all values of $f$ is called the range of $f$. – Erik G. Jan 18 '13 at 5:06
• @swanar You appear to be correct - it does not seem that Rudin gives the exact definition. That being said, it is not a good idea to get hung up on this sort of thing. Being precise is only good until it begins to interfere with actual learning. Do catch the little imprecisions Rudin makes as you read, but if the intended meaning is obvious like here (when he says function, he means function the way you mean function) then it is better to acknowledge it and move on rather than come to a complete stop. Mathematical literature is rarely as precise as your "intro to mathematics" textbook. – Gyu Eun Lee Jan 18 '13 at 5:32
• "an element of $B$" --- why do you say Rudin does not rule out one-many? I think it's pretty clear that he does. – Gerry Myerson Jan 18 '13 at 6:25
• i agree with gerry. Maybe i misinterpreted the language. When i asked someone about this, he said that a "well defined function" must not be one many but function which is not "well defined" maybe one to many. is he wrong by saying that? – swanar Jan 18 '13 at 7:22

Copying from the definition given in Rudis above: suppose that for each element $x$ of $A$ there is associated, in some manner, an element $b$ of $B$ ...

the linguistic interpretation rests on the word 'an' indicating that with each element in the domain $A$ there is associated just one (precisely one, exactly one, more than zero and less than two) element in $B$. Thus, it is to be understood that Rudin excludes multivalued functions as well as functions allowed to not attain a value at a point.

The more rigorous definition of function in terms of a relation specifying a certain condition might appeal to some since it is more rigorous but it also serves to obscure the process-like nature that one usually associates with the word 'function'. This is especially true in analysis where the subject matter is the study of the analytic properties of functions and not the set-theoretic issues of functions.

Rudin gives the following definition:

Definition 2.1 Consider two sets $A$ and $B$, whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, an element of $B$, which we denote by $f(x)$. Then $f$ is said to be a function from $A$ to $B$ (or a mapping of $A$ into $B$). The set $A$ is called the domain of $f$ (we also say $f$ is defined on $A$), and the elements $f(x)$ are called the values of $f$. The set of all values of $f$ is called the range of $f$.

In contrast:

Definition: Let $A$ and $B$ be two arbitrary sets. We say their cartesian product, denoted, $A\times{B}$, is the set $\{(a,b)\mid a\in{A}\ \mbox{and }b\in{B}\}$. That is the cartesian product is the set of all ordered pairs of elements in $A$ and $B$. A (binary) relation, $R$, is any subset of $A\times{B}$.

Thus the definitions are quite distinct. So, because I have a relation between two sets, does not imply I have a function. Whereas, a function is a special kind of relation.

See here, a function is described as functional, elements in the domain are related to distinct elements in the codomain and left-total: http://en.wikipedia.org/wiki/Relation_(mathematics)

I agree with you. Rudin should really have said, "there is associated, in some manner, a unique element $b \in B$". Without inserting this key word, the definition allows $f$ to be multivalued.

• I disagree, Rudin's definition is unambiguous (and correct). – Did Jan 18 '13 at 7:35
• @did Well what part of Rudin's definition ensures that $f(x)$ is unique? If I say, "For every cause, there is an effect", do you interpret that as meaning for a given cause there is one and only one effect? – Ink Jan 18 '13 at 7:39
• The part which says with each element x of A there is associated, in some manner, an element of B. If some elements in A had no associated element of B, one would say one or zero. If some had more than one, one would write one or more. – Did Jan 18 '13 at 7:49
• @did Are you telling me that $\forall x \exists! y$ and $\forall x \exists y$ are the same? – Ink Jan 18 '13 at 7:55
• In normal English usage the relative clause in ‘an element of $B$, which we denote by $f(x)$’ implies that the element is unique. If this is not the case in your idiolect, you’re not speaking the same language as most of us. – Brian M. Scott Jan 18 '13 at 15:19