Proving Suprenum of A and B: Bob's Question

[Bob19]

Hello

I have two non-empty sets A and B which is bounded above by R.

Then I'm tasked with proving that

$$sup(A \cup B) = max(sup A, sup B)$$

which supposedly means that $$sup(A \cup B)$$ is the largest of the two numbers sup A and sup B.

Can this then be written as $$sup(A) < sup(A \cup B)$$ and $$sup(B) < sup(A \cup B)$$ ?

Can this then be proven by showing that $$sup(A) < sup(A \cup B)$$ is true?

Or am I totally on the wrong path here??

/Bob

 

[matt grime]

correct your tex and just verify the definitions of sup.

 

[Bob19]

correct your tex and just verify the definitions of sup.

My definition of Supremum is a follows:
every non-empty, bounded above subset of R has a smallest upper bound.

Then $$sup(A \cup B)$$ has a larger smallest upper bound than sup(A) and sup(B) according to the definition of Supremum ?
Does this prove the given argument in my first post?
/Bob

p.s. If my idear is true, can this then be proven by taking a number z, which I then prove $$z \in sup(A \cup B)$$ but $$z \notin sup(A)$$ and $$z \notin sup(B)$$ ?

/Bob

 

[matt grime]

why are you treating sup as a set (and taking elements in it?). Sup is not a set, it is an element of R.

Sup of a set is the least upper bound (when it exists)

obivously the least upper bound of AuB is the max of the least upper bounds, but you need to verify it, ie show it is an upper bound, and show it is the least upper bound. The first is easy, the second slightly harder.

 

[Bob19]

obivously the least upper bound of AuB is the max of the least upper bounds, but you need to verify it, ie show it is an upper bound, and show it is the least upper bound. The first is easy, the second slightly harder.

Okay those two aspects then prove the argument that

sup(AuB) = max(sup A, sup B) ?


/Bob

 

[matt grime]

As I explained to someone else earlier tonight, I can easily see the answer because of experience, YOU need to demonstrate that you understand the answer by not having to let me fill in any blanks. If you don't see that an argument proves something then YOU need to do some work to rectify that, not me.