Undergrad Is there a difference between one to one and well defined?

[fahraynk]

if a function is one to one, the function maps every element of set A to 1 element of set B.

If a function is well defined, if $(a,b),(a,c) \in f,$ then $b=c$

These seem the same to me; do they mean the same thing?

 

[fresh_42]

well-defined: no two different images of the same point of the domain
injective (into): no two points of the domain map to the same image
surjective (onto): all points in the codomain are hit by some point
bijective (one-to-one): injective and surjective, i.e. pointwise correspondence of domain and codomain

Well-defined distinguishes relations from functions. Injectivity is a kind of embedding. Surjectivity simply means image and codomain are the same. Well-definition only gives you a function. It allows not all points of the codomain to be hit by the function, as well as it allows many points to be mapped on the same image point. Both are not allowed for a one-to-one bijection.

 

[fahraynk]

well-defined: no two different images of the same point of the domain

Not sure what you mean,
For two sets, A and B, and a function that maps A-->B
Injective means no two points in domain A map to the same point in set B
Does well defined mean: no two points in set B map to the same point in set A ?

 

[Stephen Tashi]

By the definition you quoted for "well defined", the function given by the set of ordered pairs F = { (1,3), (2,3), (3,4) } is well defined, but it is not 1-to-1. For example, if (a,b) and (a,c) are in F with a = 1 then b=c=3. And F is not 1-to-1 because it maps both 1 and 2 to 3. The definition you quoted for "well defined" just says F is actually a function -i.e. it maps a single value such as 1 to only one other value. By the usual definition of function, there is no such thing as a function that is not well defined.

 

[member 587159]

By the definition you quoted for "well defined", the function given by the set of ordered pairs F = { (1,3), (2,3), (3,4) } is well defined, but it is not 1-to-1. For example, if (a,b) and (a,c) are in F with a = 1 then b=c=3. And F is not 1-to-1 because it maps both 1 and 2 to 3. The definition you quoted for "well defined" just says F is actually a function -i.e. it maps a single value such as 1 to only one other value. By the usual definition of function, there is no such thing as a function that is not well defined.

Indeed, to expand further on this answer: In mathematics proofs, you will often see that one says, after writing something down that looks like a function (but could in theory be a relation) "This is a well defined function". Then, they just mean that what is written down is a function, so we can associate one and only one image to each original in the domain.

 

[fresh_42]

Not sure what you mean,
For two sets, A and B, and a function that maps A-->B
Injective means no two points in domain A map to the same point in set B
Does well defined mean: no two points in set B map to the same point in set A ?

No. It means no one point in A maps to two points in B. This means the inverse of squaring $x^2 \mapsto \pm{x}$ isn't well-defined, one has to choose either $+x$ or $-x$.

The examples become more interesting, if points represent entire classes of a kind, e.g. the class of all even integers $\mathcal{E}$ and the class of all odd integers $\mathcal{O}$. Then we could define $\mathcal{E} \mapsto 0\, , \,\mathcal{O} \mapsto 1$, the remainders by division by $2$. But we won't get a well-defined defined function, if we had chosen $f\, : \,\mathcal{E} \ni 2n = 2^p\cdot m \mapsto p$ because not only $2n=2^pm$ represents the class $\mathcal{E}$ but all other even numbers represent the same class. And although we have the equation for classes: $\mathcal{E} = [2n] = [2^pm]$ we also have $\mathcal{E} = [4n] = [2^{p+1}m]$ and $f:A \rightarrow B$ won't be well-defined: The same one point $\mathcal{E}$ of $A=\{\mathcal{E},\mathcal{O}\}$ would be mapped either to $p$ or to $p+1$ which are two different points in $B=\mathbb{Z}$.

 

[PeroK]

Being well defined is more fundamental. In general it says that the mathematical symbols you have written make sense. And if you declare something to be a function, say, then that something must have the properties of a function.

Examples of functions not being well defined include 1) where the function is not specified on all of the domain; 2) where the function rule cannot be carried out on all of the domain; 3) where the function value is not in the specified range;4) where the function value is ambiguous or explicitly non unique.

Note also that being well defined is not a property of a function. All functions are well defined. If something is not a well defined function, then it is not a function.

This contrasts with one-to-one, which is a property of some functions and not of others.

 

[fresh_42]

Note also that being well defined is not a property of a function.

I know what you mean and we could start a rather sophisticated debate here on what a property of an object is. Instead I want to say that this statement is likely confusing. A function is a relation with certain properties. The most important one is well-definition, as all others are a matter of taste. E.g. I would consider $f\, : \,\mathbb{R} \longrightarrow \mathbb{R} \, , \,x \mapsto +\sqrt{x}$ a function. According to your definition, its domain is $\mathbb{R}_0^+$. However, the distinction between the sets $\mathbb{R}$ and $\mathbb{R}_0^+$ makes the condition irrelevant, as you basically have to define the domain as the sets of points, where the function is defined. Otherwise there would be points allowed in $A$ which do not belong to the domain, which you ruled out.

What's left as an essential is the distinction of a function from an arbitrary relation, which is done by the requirement of well-definition. Whether to call this a property of the function, or a property of a relation, to get a function, is simply confusing and a matter of philosophy.

 

[hyunxu]

well-defined: no two different images of the same point of the domain
injective (into): no two points of the domain map to the same image
surjective (onto): all points in the codomain are hit by some point
bijective (one-to-one): injective and surjective, i.e. pointwise correspondence of domain and codomain

Actually one to one and onto is bijective and injective means one to one

 

[fresh_42]

Actually one to one and onto is bijective and injective means one to one

Some use it this way, which I find confusing. It's better to say into or name it what it is: in-, sur-, or bijective. One-to-one without hitting all elements of the codomain is sloppy in my opinion.

 

[hyunxu]

Some use it this way, which I find confusing. It's better to say into or name it what it is: in-, sur-, or bijective. One-to-one without hitting all elements of the codomain is sloppy in my opinion.

Yeah.almost I too confused remembering the names.I agree.

 

[PeroK]

I know what you mean and we could start a rather sophisticated debate here on what a property of an object is. Instead I want to say that this statement is likely confusing. A function is a relation with certain properties. The most important one is well-definition, as all others are a matter of taste. E.g. I would consider $f\, : \,\mathbb{R} \longrightarrow \mathbb{R} \, , \,x \mapsto +\sqrt{x}$ a function. According to your definition, its domain is $\mathbb{R}_0^+$. However, the distinction between the sets $\mathbb{R}$ and $\mathbb{R}_0^+$ makes the condition irrelevant, as you basically have to define the domain as the sets of points, where the function is defined. Otherwise there would be points allowed in $A$ which do not belong to the domain, which you ruled out.

What's left as an essential is the distinction of a function from an arbitrary relation, which is done by the requirement of well-definition. Whether to call this a property of the function, or a property of a relation, to get a function, is simply confusing and a matter of philosophy.

Interesting. I would never consider your example as a - strictly - well-defined function. A better example is, say, defining the inverse of a matrix. You have to change the domain from the set of all matrices.

On the other point, it's definitely more than philosophy. You cannot start a mathematical argument with something like:

"Let $f$ be a function that is not well defined ..."

 

[fresh_42]

On the other point, it's definitely more than philosophy. You cannot start a mathematical argument with something like:

"Let fff be a function that is not well defined ..."

Of course not. But whether the defining properties are related to the object, i.e. the function, or the wider range of the domain, i.e. relations, where functions are defined on, is a matter of philosophy: An animal without a trunk isn't an elephant. Now is the trunk a property of the elephant or of animals in general? This is the distinction you've made, and this is a matter of linguistics. Function includes well-definition, correct, but whether this can be called a property of functions or not, is a matter of philosophy. I think it is, but I read your statement as it wasn't.

 

[PeroK]

Of course not. But whether the defining properties are related to the object, i.e. the function, or the wider range of the domain, i.e. relations, where functions are defined on, is a matter of philosophy: An animal without a trunk isn't an elephant. Now is the trunk a property of the elephant or of animals in general? This is the distinction you've made, and this is a matter of linguistics. Function includes well-definition, correct, but whether this can be called a property of functions or not, is a matter of philosophy. I think it is, but I read your statement as it wasn't.

You can certainly consider a relation that is not necessarily a function. But, a function - by definition, not philosophy - must have all the defining properties of a function.

In fact, it's only linguistically that "well defined" appears to have the nature of a mathematical property. If you look at it mathematically, you can see that it's not a property.

Mathematically, "well defined function" is logically equivalent to "function".

 

[fresh_42]

You can certainly consider a relation that is not necessarily a function. But, a function - by definition, not philosophy - must have all the defining properties of a function.

Yes, but you've said that well-definition isn't a property of a function!

Note also that being well defined is not a property of a function.

It cannot both be true and that is what I said is confusing: the change of levels between to what the quality property is related to: the relation as the defining set for functions, or the function itself. And this has nothing to do with mathematics.

 

[PeroK]

Yes, but you've said that well-definition isn't a property of a function!

It cannot both be true and that is what I said is confusing: the change of levels between to what the quality property is related to: the relation as the defining set for functions, or the function itself. And this has nothing to do with mathematics.

Is it fair to say the following:

My definition of a mathematical thing would be:

Property A
Property B
Property C

And your definition would be:

Property X: It is well defined
Property A
Property B
Property C

I.e. All mathematical objects have a generic property of being well-defined.

 

[jbriggs444]

I would say that it is a use/reference distinction. Being "well defined" is not a property of a mathematical object. It is a property of a description that purports to define a mathematical object. An object is well defined by a description if that description uniquely identifies that mathematical object.

[Up to isomorphism or whatever level of identity one is interested in]

 

[fresh_42]

I would say that it is a use/reference distinction. Being "well defined" is not a property of a mathematical object. It is a property of a description that purports to define a mathematical object. An object is well defined by a description if that description uniquely identifies that mathematical object.

[Up to isomorphism or whatever level of identity one is interested in]

Yes, the trouble is the same wording (property) for what defines a relation to be a function, and what behavior the defined object, the function has. This is already seeded in the question itself: "difference between one to one" [property of the function as a function] "and well defined" [property of a relation to be a function]. By calling both properties of a function, we already mixed the two meanings. My point has been less this difference of levels, but more the practical point of view: In order to define a function, well-definition is necessary to check as it is the defining property. Whether the resulting function has properties like bijectivity is then another question on a different level. But practically proofs are usually structured like this:

• rule how the mapping has to be performed
• well-definition to guarantee a function
• surjectivity (one half of the way towards a bijection)
• injectivity (the other half)

which means, that these levels are practically not distinguished. The first two are needed and necessary to be able to speak of functions, the last two are properties of the result which might hold or not.