Register to reply

What is the meaning of that a function F is well-defiened ?

by Maths Lover
Tags: function, meaning, welldefiened
Share this thread:
Maths Lover
#1
Nov7-12, 03:30 PM
P: 66
what does well defiened mean ?

how can we check that a function is well defiened ?
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
arildno
#2
Nov7-12, 03:38 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
That a single input value yields a single output value.
micromass
#3
Nov7-12, 03:40 PM
Mentor
micromass's Avatar
P: 18,323
Let me give an example. Let [itex]\mathbb{Q}[/itex] be the rational numbers. Let's define the following "function":

[tex]f\left(\frac{m}{n}\right)=\frac{m+1}{n+1}[/tex]

At the first sight, there is not really a problem. But let's look deeper. We have

[tex]f\left(\frac{1}{2}\right)=\frac{2}{3}[/tex]

and

[tex]f\left(\frac{2}{4}\right)=\frac{3}{5}[/tex]

But, as we know, the numbers 1/2 and 2/4 are equal, but the numbers 2/3 and 3/5 are not equal. So the function f sends 1/2 to two different values: 2/3 and 3/5.
However, a function is defined as sending a value in the domain to a UNIQUE value in the codomain. Here, we have sent 1/2 to two different numbers, which means that f is not a function. We usually say that ''f is not well-defined'' (I don't think the expression is really formally correct, but it is used everywhere).

Another way a function could not be well-defined is that it send something to a value not in the codomain. For example: [itex]f(x)=\sqrt{x}[/itex] is not well-defined if the domain and codomain are both [itex]\mathbb{R}[/itex]. Indeed, [itex]\sqrt{-1}[/itex] is not in the codomain.

So, if you are given a ''function'', you should always check if a value is being sent to a single other value, and not to multiple values. Furthermore, the value should be in the codomain. Knowing when this is a problem requires a bit of experience, but the problem arises usually in things like ''quotients''.


Register to reply

Related Discussions
The meaning of Translational partition function General Physics 0
What is the meaning by a function with continuous 1st and 2nd derivatives? Differential Equations 10
Singularity Function Meaning General Engineering 0
Meaning of Zeros of Partition function Atomic, Solid State, Comp. Physics 9
Meaning of the s-domain absolute value function Electrical Engineering 21