Can Multiple Outputs from a Single Input Help in Constructing Non-Linear Curves?

In summary, the conversation discusses the possibility of encapsulating the behavior of an object in the object itself, as well as the concept of obtaining multiple outputs from a single input in mathematical terms. It is suggested that this can be achieved through a relation between the set of input values and the set of output values, rather than a traditional function. The conversation also explores the idea of an input yielding infinite outputs and the use of mathematical terminology to clarify the discussion.
  • #1
pairofstrings
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<Moderator's note: This is a spin-off from another thread.>

Svein said:
How are you going to find the area and how do you know that the area is the correct one?
I will find out axioms to find out an answer to a question - axioms guarantees that my solution to a mathematical problem is correct.

I have another question: A function 'y' in 'x' yields a single value as output on an input. Is there any mathematical construct that let's me acquire multiple output on single input so that I could talk about non-linear curves - I think, obtaining multiple points at once from this hypothetical mathematical construct will let me build any kind of non-linear curve.
 
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  • #2
pairofstrings said:
I have another question: A function 'y' in 'x' yields a single value as output on an input. Is there any mathematical construct that let's me acquire multiple output on single input so that I could talk about non-linear curves - I think, obtaining multiple points at once from this hypothetical mathematical construct will let me build any kind of non-linear curve.
A general curve in the XY plane can be defined by parameterizing it with another variable. You can define the points (x(t), y(t)), t ∈ [0,1] to parameterize the curve using the parameter t. There can be multiple y values for the same x value.
A familiar example is the parameterization of a circle: (cos(θ), sin(θ)), 0≤θ≤2π.
 
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  • #3
pairofstrings said:
I will find out axioms to find out an answer to a question - axioms guarantees that my solution to a mathematical problem is correct.
Sounds as if you have some basic misconceptions of what axioms are. You just said: I define an answer to a question to be the answer to the question. Imagine the following question: "What is the solution to ##x\cdot a = a## for any ##a\,##? Let us call it ##x=2\,##." You can do this, but then you cannot use ##2## anymore as the result of ##1+1##. You need consistency, not merely an arbitrary definition.
I have another question: A function 'y' in 'x' yields a single value as output on an input. Is there any mathematical construct that let's me acquire multiple output on single input ...
This is called a relation. E.g. ##x \leftrightarrow \{a\,\vert \,a^2=x\}## or ##\mathcal{R}=\{(x,a)\,\vert \,a^2=x\}\,##.
... so that I could talk about non-linear curves - I think, obtaining multiple points at once from this hypothetical mathematical construct will let me build any kind of non-linear curve.
I cannot see what one has to do with the other. Of course does any scribbling of a curve in the plane define a relation, but this view isn't very useful in terms of applications.
 
  • #4
I want to talk about my first post of this thread in little detail.

Is it possible to encapsulate behavior of an object in the object itself - assume that the object is at initial point, assume that there are multiple objects and these objects have behavior embedded into themselves? How do I say this in mathematical terms or how do talk about his mathematically? I want to talk about such objects collectively and independently.

y = how this expression could be?
y is equal to what?
separation_zpssmhrzce3.jpg

The above is a single pseudo equation talking about multiple objects collectively - I am trying to write a single equation that talks about two objects collectively.

The following equation yields one result, not two.
multiple-y_zpsnhonksvq.jpg


I need two values from a single input. So, what is it that separates these two objects and yet be the same 'y = '?
How do I write 'y = ' for this? Is it possible to write 'y = ' for the system that yields multiple outputs for a single input?
Is this Relation - the Relation helps to connect multiple objects to form a single 'y = '?
If yes, then what is the symbol that can isolate multiple objects in this single 'y = ' ?

What I am trying to find is that I want single 'y = ' for multiple objects.
y_zpsxercjgvl.jpg


A single input may also yield infinite outputs. I can plot a graph for a single input (on X-axis) against multiple or infinite output (on Y-axis)

many-y_zpsrhqeh65p.jpg


I want an input to bifurcate into multiple points or infinite points or 'n' points.

The above picture is the gist of this entire discussion. I hope I was clear in presenting the question.
 

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  • #5
pairofstrings said:
Is it possible to encapsulate behavior of an object in the object itself - assume that the object is at initial point, assume that there are multiple objects and these objects have behavior embedded into themselves? How do I say this in mathematical terms or how do talk about his mathematically? I want to talk about such objects collectively and independently.
I don't think this is possible. A mathematical equation involves some attribute of an object, such as its position, its velocity, its mass, etc., but doesn't describe the object itself.

pairofstrings said:
y = how this expression could be?
y is equal to what?
Here's an example of one possibility: ##y = \pm x##. This is really shorthand for ##y = x \text{ or } y = -x##. For a given x value, there are two y values. Note that this is a relation between the set of x values and the set of y values. It is not a function.

pairofstrings said:
I need two values from a single input. So, what is it that separates these two objects and yet be the same 'y = '?
How do I write 'y = ' for this?
 
  • #6
pairofstrings said:
A single input may also yield infinite outputs. I can plot a graph for a single input (on X-axis) against multiple or infinite output (on Y-axis)

You aren't asking a specific question. Since you don't use mathematical terminology, let's try the terminology of computer programming. What is your definition of an "input"? What kind of data structure is it? Must it be a floating point number? What kind of data structure is the output Y?
 
  • #7
many-y_zpsrhqeh65p.jpg


1.
This is the curve that I found when I traced the surface of an object.
The curve is non-linear. Is there a way to find the equation of the curve?
I want to be able to recreate the curve 'y' when I substitute Integers into 'x'.
'y' is a function in 'x'.

2.
I can count objects (square, triangle, circle, rectangle, etc) that were used to build a particular object. What I want is that the count should also describe how the particular object visually appears.
 

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  • #8
pairofstrings said:
View attachment 219266

1.
This is the curve that I found when I traced the surface of an object.
The curve is non-linear. Is there a way to find the equation of the curve?
I want to be able to recreate the curve 'y' when I substitute Integers into 'x'.
'y' is a function in 'x'.
You can approximate the equation of the curve using curve fitting, about which there are a number of techniques. See https://en.wikipedia.org/wiki/Curve_fitting
The inputs to this equation don't have to be integers.
pairofstrings said:
2.
I can count objects (square, triangle, circle, rectangle, etc) that were used to build a particular object. What I want is that the count should also describe how the particular object visually appears.
This seems unrelated to your question in post #1. I suppose you could describe an object with a number of small cubes, keeping track of the position and size of one dimension of each cube.
 

Related to Can Multiple Outputs from a Single Input Help in Constructing Non-Linear Curves?

What are functions and relations?

Functions and relations are mathematical concepts that describe how one quantity (the dependent variable) changes in response to changes in another quantity (the independent variable).

What is the difference between a function and a relation?

A function is a type of relation where each input (x-value) has exactly one output (y-value). In other words, for every x-value, there is only one y-value. A relation, on the other hand, can have multiple outputs for a single input.

How do you determine if a graph represents a function?

To determine if a graph represents a function, you can use the vertical line test. If a vertical line can be drawn through any point on the graph and only intersects the graph at that point, then it is a function. If the vertical line intersects the graph at more than one point, then it is not a function.

What is the difference between a linear and a nonlinear function?

A linear function is a function that has a constant rate of change, meaning that the output changes by the same amount for every unit change in the input. On the other hand, a nonlinear function does not have a constant rate of change, meaning that the output changes by different amounts for different unit changes in the input.

How can you represent a function algebraically?

A function can be represented algebraically using a function rule, where the input (x-value) is plugged into the rule to determine the output (y-value). For example, the function rule y = 2x + 3 represents a linear function where the output is equal to two times the input plus three.

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