MHB Partial Order Relation on a Functions Set

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The discussion centers on understanding a partial order relation R defined on a set of functions from [0,1] to [0,infinity), where fRg if f(x) ≤ g(x) for all x in [0,1]. Participants explore whether specific statements about the existence of largest and smallest elements, maximal and minimal elements, and comparability of functions are true or false. It is clarified that most functions will not be comparable, as they may satisfy f(x) < g(x) for some x and g(x) < f(x) for others. The smallest function in this context is identified as f(x) = 0. The conversation highlights the complexities of ordering functions compared to simpler numerical examples.
Yankel
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Hello,

I have a question which includes several statements, which I need to decide if they are true or false. I am not sure how to do it, if you could give me hints or "leads", it will mostly appreciated.

R is a partial order relation on A, a set of functions from [0,1] to [0,infinity) such that fRg if and only if f(x)<=g(x) for all x which belongs to [0,1].

In this relation:

1) There is a largest and smallest element
2) There is no largest element but there is a smallest one
3) There is more than one maximal element
4) There is more than one minimal element
5) Every 2 elements of A are comparable

I thought maybe to try and build a Hasse diagram, but unlike simple example with pairs of numbers, here I found it more difficult.

How do I compare two functions and order them ?

Thank you.
 
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Yankel said:
Hello,

I have a question which includes several statements, which I need to decide if they are true or false. I am not sure how to do it, if you could give me hints or "leads", it will mostly appreciated.

R is a partial order relation on A, a set of functions from [0,1] to [0,infinity) such that fRg if and only if f(x)<=g(x) for all x which belongs to [0,1].

In this relation:

1) There is a largest and smallest element
2) There is no largest element but there is a smallest one
3) There is more than one maximal element
4) There is more than one minimal element
5) Every 2 elements of A are comparable

I thought maybe to try and build a Hasse diagram, but unlike simple example with pairs of numbers, here I found it more difficult.

How do I compare two functions and order them ?

Thank you.
You are told how to "compare two functions and order them" in the definition of the order relation: given two functions f and g, f< g if and only if f(x)< g(x) for all x. Of course, most functions will NOT be "comparable" because we will have f(x)< g(x) for some x and g(x)< f(x) for others.
 
I see. Which function is the smallest in this case ? Is it f(x)=0 ?
 
Yankel said:
Which function is the smallest in this case ? Is it f(x)=0 ?
Yes.
 

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