What is the exact definition of a function?

  • Context: Undergrad 
  • Thread starter Thread starter woundedtiger4
  • Start date Start date
  • Tags Tags
    Definition Function
Click For Summary

Discussion Overview

The discussion centers around the definition of a function in mathematics, exploring various interpretations and formal definitions. Participants reference different texts and definitions, examining the implications of these definitions on the understanding of functions, including their properties related to inputs and outputs.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant cites Spivak's definition of a function as a collection of pairs where each input has a unique output, while another text by Kolmogorov suggests that an output can correspond to multiple inputs, leading to confusion about the definitions.
  • Another participant clarifies that the first definition emphasizes unique outputs for each input, while the second allows for multiple inputs to yield the same output, using the example of the function f(x)=x^2.
  • A formal definition of a function is presented as a set of ordered pairs where no two pairs have the same first member.
  • One participant discusses the vertical line test as a method to determine if a graph represents a function, although another participant points out that this test is limited to graphical representations.
  • Concerns are raised about defining a function as a "rule," with one participant arguing that this definition is inadequate for more complex functions, especially when considering uncountable sets.
  • Another participant reflects on their initial confusion and suggests that the definitions from Spivak and Kolmogorov focus on different aspects of functions, specifically range and domain.
  • A question is posed regarding a set of pairs that includes duplicates, leading to a clarification that in set theory, duplicates are not counted, reinforcing the uniqueness requirement of the first member in function definitions.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of functions, with no consensus reached on a singular definition. Some participants agree on the formal definition, while others challenge the sufficiency of defining functions in terms of rules or graphical representations.

Contextual Notes

Participants highlight limitations in definitions, such as the dependence on context (e.g., graphical vs. set-theoretic definitions) and the implications of defining functions in terms of rules versus ordered pairs.

woundedtiger4
Messages
188
Reaction score
0
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:
Physics news on Phys.org
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 into 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 bothEDIT: 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
 
Hello,

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.
 
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.
 
binomial said:
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.



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.

DonAntonio
 
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.
 
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)).
 
mathwonk said:
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)).

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.
 
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)
 
  • #10
hivesaeed4 said:
Hello,

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.

I'm not a mathematician so forgive me for asking a dumb question; what if the function is given as:

(1,10)
(2,12)
(2,12)
(3,14)

Edit: I get it- no two that have *only* the same first member?
 
  • #11
Antiphon said:
I'm not a mathematician so forgive me for asking a dumb question; what if the function is given as:

(1,10)
(2,12)
(2,12)
(3,14)

Edit: I get it- no two that have *only* the same first member?

Technically, a function is a set of ordered pairs; as a set of ordered pairs, you count (2,12) only once.
 

Similar threads

Undergrad How are the following three definitions subtly different?
  • · Replies 15 ·
Replies
15
Views
2K
Graduate Definitions of Cylinder Sets and Cylinder Set Measure
  • · Replies 2 ·
Replies
2
Views
5K
Undergrad Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Showing equivalence of two definitions of essential supremum
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Uniform closure of algebra of bounded functions is uniformly closed
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Confused by proof in Hocking and Young
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Local Basis in Topology .... Definitions by Croom and Singh .... ....
  • · Replies 14 ·
Replies
14
Views
2K
Graduate Definitions of Lebesgue integral
  • · Replies 11 ·
Replies
11
Views
3K
MHB Understanding Riemann Integrable Functions: Interpreting D&K Pages 427-428
  • · Replies 2 ·
Replies
2
Views
2K