What is the ambiguity in forming new functions?

[DumpmeAdrenaline]

Why do we want to always deal with single valued functions?
In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range.

The first definition does not negate the assignment of one element to two or more different elements from being a function. If we employ the first definition for a function, we can still talk about 1-1 and onto functions.

However, there is ambiguity in forming new functions from old functions.
If an element in the domain is mapped into two different images, which of the two images do we choose to take on to perform the allowed operations on functions.

 

[anuttarasammyak]

Why do we want to always deal with single valued functions?

Because it is simple and easy, I think. As an example inverse function of y=x^2 is actually
$$x=\pm \sqrt{y},y>0$$
Here we prefer to choose positive one and invent a symbol $\sqrt{}$ for it. Then we will proceed to deal multi valued-functions in an advanced curriculum.

 

[PeroK]

If an element in the domain is mapped into two different images, which of the two images do we choose to take on to perform the allowed operations on functions.

If you have a multi-valued function, then you must be careful how you operate on the set of function values. You'll see things like:
$$x = -b \pm \sqrt{b^2 - 4c}$$Which can also be written $$x = -b + \sqrt{b^2 - 4c} \ \text{or} \ x = -b - \sqrt{b^2 - 4c}$$Here, technically, $x(b,c)$ is a multi-valued function of $b$ and $c$. And, you always have to take into account that there are two possible values for $x$.

 

[Sunay_]

I think the idea is to be able to analyze mathematical objects using calculus. If you take derivative of some function and it gives two different results how will you interpret it? In order to use calculus you have to have well-defined functions.

 

[Mark44]

Why do we want to always deal with single valued functions?

We don't always do this, but as already mentioned, single-valued functions are simpler. Further along in mathematics you'll come to vector-valued functions -- functions whose output is a vector.

For example, $\vec r(t) = <f(t), g(t), h(t)>$.

 

[pasmith]

Why do we want to always deal with single valued functions?
In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range.

The first definition does not negate the assignment of one element to two or more different elements from being a function.

I disagree. In fact I think the first definition is just a sepcial case of the second, applying only to "numbers". But in both cases the central idea is that for each $x$ there is exactly one $y$ such that $f(x) = y$

 

[Office_Shredder]

If you want arbitrary multi value functions $\mathbb{R}\to \mathbb{R}$, this is the same as single valued functions $\mathbb{R}\to P(\mathbb{R})$ where $P(\mathbb{R})$ is the power set of $\mathbb{R}$, the set of all subsets of the real numbers.

But these functions can be very strange. You might have some f where $f(0)= \mathbb{Q}$, $f(1)= \{0,\pi,12\}$, and $f(2)$ is the set of transcendental numbers.

Ok, what do you do with this function? It seems kind of useless. Even among single valued functions, we restrict ourselves to continuous functions, or differentiable functions, or integrable functions, in order to do useful mathematics.

 

[PeroK]

Perhaps at this point we should wait for the OP to respond.

 

[Stephen Tashi]

The first definition does not negate the assignment of one element to two or more different elements from being a function. If we employ the first definition for a function, we can still talk about 1-1 and onto functions.

Where did you get the first definition? The usual definition of a real valued function does not allow it assign one number to two distinct numbers.

 

[DumpmeAdrenaline]

Because it is simple and easy, I think. As an example inverse function of y=x^2 is actually
$$x=\pm \sqrt{y},y>0$$
Here we prefer to choose positive one and invent a symbol $\sqrt{}$ for it. Then we will proceed to deal multi valued-functions in an advanced curriculum.

I can imagine why we prefer to choose the positive one, as we can define for instance the distance between two numbers in terms of the positive square root. Some operations cannot be interpreted with multiple images so without loss of generality we deal with single valued functions.

I disagree. In fact I think the first definition is just a sepcial case of the second, applying only to "numbers". But in both cases the central idea is that for each $x$ there is exactly one $y$ such that $f(x) = y$

The second definition applies to all well defined sets with objective tests for membership (an element does not necessarily mean we are only dealing with real numbers).

If you want arbitrary multi value functions $\mathbb{R}\to \mathbb{R}$, this is the same as single valued functions $\mathbb{R}\to P(\mathbb{R})$ where $P(\mathbb{R})$ is the power set of $\mathbb{R}$, the set of all subsets of the real numbers.

But these functions can be very strange. You might have some f where $f(0)= \mathbb{Q}$, $f(1)= \{0,\pi,12\}$, and $f(2)$ is the set of transcendental numbers.

Ok, what do you do with this function? It seems kind of useless. Even among single valued functions, we restrict ourselves to continuous functions, or differentiable functions, or integrable functions, in order to do useful mathematics.

I can infer from this that we limit ourselves to the study of physical situations where for every (one input or more) we obtain a unique value for y in the image set.

 

[Office_Shredder]

I can infer from this that we limit ourselves to the study of physical situations where for every (one input or more) we obtain a unique value for y in the image set.

I think the point is more, we limit ourselves to the study of mathematical objects with sufficient structure to say interesting things. Often these structures are inspired by physics, but not always (e.g. number theory). If you permit arbitrary multi valued functions where the number of values per input is arbitrary, then you just have arbitrary functions $\mathbb{R}\to P(\mathbb{R})$. From a set theory perspective this is maybe interesting - e.g. what is the cardinality of the set of all such functions? But in terms of asking 'do these functions have any similar properties beyond the one that we wrote down' the answer is no. Versus e.g. single valued continuous functions, they have lots of shared properties - they satisfy the intermediate value theorem, they achieve their minimum and maximum value on any closed interval, etc. There are lots of things you can prove about a function once you know it's continuous, and proving things is the goal of mathematics.

 

[valenumr]

Are you asking about functions of multiple variables, e.g. f(x,y,z)? Those would still be functions.

 

[Mark44]

Are you asking about functions of multiple variables, e.g. f(x,y,z)? Those would still be functions.

No, That's not what this thread is about.

From post #1:

Why do we want to always deal with single valued functions?

 

[Ssnow]

In mathematics you are able to differentiate single valued-functions and multi-valued functions.

https://en.wikipedia.org/wiki/Multivalued_function

Ssnow

 

[sophiecentaur]

The usual definition of a real valued function does not allow it assign one number to two distinct numbers.

By that definition, the "real valued functions" are a small minority of functions and not much use to us. Afaics, it implies a monotonic relationship and you find a lot of those in real life that don't.
I did my formal Mathematical Analysis course more than fifty years ago so things may have changed by now, but still . . . . .

 

[PeroK]

By that definition, the "real valued functions" are a small minority of functions and not much use to us. Afaics, it implies a monotonic relationship and you find a lot of those in real life that don't.
I did my formal Mathematical Analysis course more than fifty years ago so things may have changed by now, but still . . . . .

I think he meant "does not allow it to map one number to two distinct numbers". Not that a function must be one-to-one.

 

[sophiecentaur]

I think he meant "does not allow it to map one number to two distinct numbers". Not that a function must be one-to-one.

So that excludes x= Root y?

 

[PeroK]

So that excludes x= Root y?

No. $f(x) = \sqrt x$ is a function. But, $f(x) = \pm\sqrt x$ is two functions or a multi-valued function.

 

[sophiecentaur]

No. $f(x) = \sqrt x$ is a function. But, $f(x) = \pm\sqrt x$ is two functions or a multi-valued function.

OOOO KKKKK you Mathematicians live in a funny old world and you have a need to tie things up tidily. So what you are saying is that most functions are, in fact 'multivalued functions'?? Fair enough

 

[PeroK]

OOOO KKKKK you Mathematicians live in a funny old world and you have a need to tie things up tidily. So what you are saying is that most functions are, in fact 'multivalued functions'?? Fair enough

Define "most"!

 

[PeroK]

OOOO KKKKK you Mathematicians live in a funny old world and you have a need to tie things up tidily. So what you are saying is that most functions are, in fact 'multivalued functions'?? Fair enough

Strictly speaking a "multi-valued function" is not a function that has the property of being multi-valued. It's a new object that is not a function.

 

[sophiecentaur]

Define "most"!

Any 'relationship' between two variables that involves a turning value. All my Maths notes that contain the word 'function' are possibly bo****x then? I used to write sentences like "any function that is continuous and differentiable in the range 0<x<1 ". Clearly not enough. I will have to tread much more carefully now; no more BS in my future posts.

 

[PeroK]

Any 'relationship' between two variables that involves a turning value. All my Maths notes that contain the word 'function' are possibly bo****x then? I used to write sentences like "any function that is continuous and differentiable in the range 0<x<1 ". Clearly not enough. I will have to tread much more carefully now; no more BS in my future posts.

The most general definition of a real-valued function on $\mathbb R$ or some subset of $\mathbb R$ (the domain of the function) is a rule that maps every real number in the domain to a real number.

Functions may have properties like being integrable, continuous, differentiable, infinitely differentiable.

Being multi-valued is not then a property, as it is excluded by definition.

A multi-valued function is a rule that maps every real number in the domain to one or more real numbers (to a set of numbers). A function is then a special case of a multi-valued function.

 

[Mark44]

So that excludes x= Root y?

I can't count the times that members here have held the misconception that $\sqrt y$ has two different values -- one positive and one negative. By definition and convention, the symbol $\sqrt y$ is the principal square root, and is the positive number x such that $x^2 = y$.

Any 'relationship' between two variables that involves a turning value. All my Maths notes that contain the word 'function' are possibly bo****x then?

Not sure what you mean by "turning value," unless possibly you mean a local minimum or local maximum point. There's a very simple test for determining whether the graph of a relation is a function or not: a vertical line never intersects more than one point on the graph. Algebraically this is saying that if f(x) = y₁ and f(x) = y₂, for distinct y₁ and y₂, and for any x in the domain of f, then f is not a function.

I used to write sentences like "any function that is continuous and differentiable in the range 0<x<1 ". Clearly not enough. I will have to tread much more carefully now; no more BS in my future posts.

No problem with the phrase above other than a possible misconception about the definition of the term, function.

 

[Stephen Tashi]

A multi-valued function is a rule that maps every real number in the domain to one or more real numbers (to a set of numbers). A function is then a special case of a multi-valued function.

A multi-valued function is to a function as a counterfeit currency is to a currency.

 

[Mark44]

A multi-valued function is to a function as a counterfeit currency is to a currency.

Not so, if you consider multi-valued functions to be synonymous with vector-values functions.

 

[wrobel]

To any multivalued mapping $f:X\to Y$ you can easily put in correspondence a single valued mapping: $\tilde f:X\to 2^Y,\quad \tilde f(x)=\{f(x)\}$ :)

 

[Office_Shredder]

Not so, if you consider multi-valued functions to be synonymous with vector-values functions.

This assumes that every input has the same number of outputs.

 

[Mark44]

This assumes that every input has the same number of outputs.

For vector-valued functions, this is the case, assuming the input value is in the domain.