Partial Order Relation on a Functions Set

[Yankel]

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.

 

[HallsofIvy]

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.

 

[Yankel]

I see. Which function is the smallest in this case ? Is it f(x)=0 ?

 

[Evgeny.Makarov]

Which function is the smallest in this case ? Is it f(x)=0 ?

Yes.