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What is the exact definition of a function?

  1. Apr 18, 2012 #1
    In text (Spivak) it says that a function is a collection of pairs of numbers with the following property: if (a,b) & (a,c) are both in the collection, then b=c; in other words, the collection must not contain two different pairs with the same first element. Now in an other text (Kolmogorov) I found that if b is a range of the function then it could have several preimages or no preimages at all (preimage means the domain) now these two definitions are opposite to each other & I don't really know that which one is true :|

    P.S. Just figured out that both definitions are same, please ignore my post
    Last edited: Apr 18, 2012
  2. jcsd
  3. Apr 19, 2012 #2
    I think you've just gotten mixed up with the second definition.

    In the first definition, the author is saying that for any given element that you input in to a function, there is one unique element as output. (we can use the vertical line test to examine this property). That's why in the example it said, if (a,b) and (a,c) are part of the range then b = c, because there is only one unique mapping for a. Now, the second definition you mentioned, the author is saying that the reverse isn't necessarily true - a function *can* have an element in the range (output) that has more than one elements as its input (preimage). To use a similar "definition", the author could say (a,b) and (x,b) does NOT imply (a=x). (imagine the function f(x)=x^2, you can plug in 2 and -2 and each of these produce 4). Do you see the difference? In that first definition we are talking about the fact that each element of the domain gets mapped to a unique element in the range. in the second definition we are talking about the fact that multiple elements in the domain can map to the same element in the range.

    Hope that helps! I think you just got a little thrown off because it sounds like the same thing is being talked about in both

    EDIT: Just saw that you said ignore you post, but wanted to point out that both definitions are NOT the same, they are talking about different things. Just wanted to make sure you were clear on the definitions the author gave
  4. Apr 19, 2012 #3

    Here is the formal definition of a function:

    A function f is defined as a set of ordered pairs (x, y) no two of which have the same first member.

    Hope that helps.
  5. Apr 19, 2012 #4
    basically if you evaluate the equation and find that it has two of the same x values, then it's not a function. just apply the vertical line test. if that line intersects the curve more than once, it's not a function. pretty much all there is to it.
  6. Apr 19, 2012 #5

    That didn't make much sense, did it? "To evaluate the function and then find... it is not a function"?

    The vertical line test may work for functions with graphs and only on a plane. There are plenty of functions which haven't such a representation.

    Hivesaeed4 already gave the formal definition from set theory.

  7. Apr 19, 2012 #6
    Lol. I said if you evaluate the "equation" - whatever it may be. I did not say if you evaluate the function. I almost said that in my initial response, though. Ha ha.
  8. Apr 20, 2012 #7


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    I f you want to think about functions, not many people think of them this way. I.e. technically this is the definition of the "graph" of a function. I.e. if A and B are sets, then any subset of the cartesian product AxB is called a correspondence between A and B. If the subset passes the vertical line test, it is the graph of a function defined on that subset (called the domain) of A consisting of the elements that occur as first entry of a point on the graph.

    The function itself is thought of as a mapping that picks up each element x of the domain from A and deposits it on top of the unique element y of B that occurs as the second entry of the unique pair having x as first entry.

    Thus many people define a function f from A to B as a rule that assigns, to each element x of A, a unique corresponding element f(x) in B. Then the graph of f is the subset of AxB consisting of all pairs (x,f(x)).
  9. Apr 20, 2012 #8
    Very problematic to define a function as a rule. If by "rule" you mean a finite string of symbols from a countable alphabet, there are only countably many rules. But there are uncountably many functions from one infinite set to another, or even from the natural numbers to the finite set {0,1}.

    The OP already gave the mathematically correct definition. The "rule" definition is only suitable for the most casual of contexts, for example presenting functions to high school students for the first time.
  10. Apr 20, 2012 #9
    Thanks everyone. Actually, I figured out my confusion after starting this thread therefore immediately I posted the 'P.S' :)
    Actually, Spivak has done the analysis on range whereas Kolmogorov on domain (Please correct me if I am wrong)
  11. Apr 20, 2012 #10
    I'm not a mathematician so forgive me for asking a dumb question; what if the function is given as:


    Edit: I get it- no two that have *only* the same first member?
  12. Apr 20, 2012 #11
    Technically, a function is a set of ordered pairs; as a set of ordered pairs, you count (2,12) only once.
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