Least upper bound/ greatest lower bound proof

In summary, the goal of the homework statement is to prove that every element of B is a lower bound for U, and that if x is the greatest lower bound of U, then x is the least upper bound of B.
  • #1
Syrus
214
0

Homework Statement


Suppose R is a partial order on A and B ⊆ A. Let U be the set of all upper bounds for B.
a) Prove that every element of B is a lower bound for U.
b) Prove that if x is the greatest lower bound of U, then x is the least upper bound of B.

Homework Equations



The Attempt at a Solution


I have proven a)

Let b ∈ B. Let u ∈ U. Then by the definition of upper bound, (b,u) ∈ R.


For the proof of b), you obviously assume the antecedent of the statement to be shown. I can't, however, seem to make out the logical form of this. Also, it appears you must use part a) in the proof of part b) as well. I have an intuitive understanding of the idea of the statement to be proven, I am just having a hard time (in-)formalizing it.

The logical form of the goal- i would think- is (∀b ∈ B)((b,x) ∈ R) and (∀u ∈ U)((x,u) ∈ R); I assume this means that x is an upper bound of B AND that x is the least upper bound of B (hence, x is the least upper bound of B).

So naturally, my proof thus far is:
Suppose x is the greatest lower bound of U. Let b ∈ B...
 
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  • #2
Syrus said:

Homework Statement


Suppose R is a partial order on A and B ⊆ A. Let U be the set of all upper bounds for B.
a) Prove that every element of B is a lower bound for U.
b) Prove that if x is the greatest lower bound of U, then x is the least upper bound of B.

Homework Equations



The Attempt at a Solution


I have proven a)

Let b ∈ B. Let u ∈ U. Then by the definition of upper bound, (b,u) ∈ R.


For the proof of b), you obviously assume the antecedent of the statement to be shown. I can't, however, seem to make out the logical form of this. Also, it appears you must use part a) in the proof of part b) as well. I have an intuitive understanding of the idea of the statement to be proven, I am just having a hard time (in-)formalizing it.

The logical form of the goal- i would think- is (∀b ∈ B)((b,x) ∈ R) and (∀u ∈ U)((x,u) ∈ R); I assume this means that x is an upper bound of B AND that x is the least upper bound of B (hence, x is the least upper bound of B).

So naturally, my proof thus far is:
Suppose x is the greatest lower bound of U. Let b ∈ B...

You've proven the part a rightly but It's not complete I guess. to prove a, remember that U is the set of all upper bounds of B which means for any b in B we have: (b,u)∈R (u in U). now let u be any element of U and fix b. that means u is bounded by an element of b. now let b vary, that means every element of B is a lower bound for you and that means B is the set of lower bounds of U.

to prove the part b, It's easy to show if x is the glb of U then x is an upper bound of B. what you need to prove is that x is also the least upper bound. that means there exists no other element like y s.t y is an upper bound of B and (y,x)∈R. It's easy to show that, I leave the proof to you. prove it by reductio ad absurdum (indirect argument).
 
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  • #3
Coulnd't there also be lower bounds which are not in B? Part of my trouble is specifying a set containing a lower bound of U, since b need not be in B.
 
  • #4
Syrus said:
Coulnd't there also be lower bounds which are not in B? Part of my trouble is specifying a set containing a lower bound of U, since b need not be in B.

Maybe there are lower bounds of U that are not in B. I guess that might happen when B is left-bounded and I'm not 100% sure about that. but in our case, that doesn't matter. because if m is a lower bound for U that is not in B and we have: [itex]\forall[/itex]b[itex]\forall[/itex]u: b<l<u then that leads us to a contradiction. that's all we need to know to prove that glb(U)=lub(B). I don't think we need anything else to prove the problem.
 
  • #5
AdrianZ said:
You've proven the part a rightly but It's not complete I guess. to prove a, remember that U is the set of all upper bounds of B which means for any b in B we have: (b,u)∈R (u in U). now let u be any element of U and fix b. that means u is bounded by an element of b. now let b vary, that means every element of B is a lower bound for you and that means B is the set of lower bounds of U.

But then B is not "the set of lower bounds of U" rather, A set of lower bounds of U.

Another thing... i am struggling with how to show that x is an upper bound of B... Even though x is the glb of U doesn't necesserily mean it is an element of U. Still somewhat confused i guess.
 

1. What is the least upper bound/greatest lower bound?

In mathematics, the least upper bound or supremum is the smallest number that is greater than or equal to all the elements in a set. Similarly, the greatest lower bound or infimum is the largest number that is less than or equal to all the elements in a set.

2. Why is the least upper bound/greatest lower bound important?

The least upper bound/greatest lower bound is important because it helps us determine the boundaries of a set. It allows us to find the smallest or largest possible value in a set, which is useful in many mathematical proofs and applications.

3. How is the least upper bound/greatest lower bound proved?

The least upper bound/greatest lower bound can be proved using the completeness axiom in real analysis. This axiom states that every non-empty set of real numbers that is bounded above has a least upper bound, and every non-empty set of real numbers that is bounded below has a greatest lower bound.

4. Can the least upper bound/greatest lower bound exist outside of real numbers?

No, the concept of least upper bound/greatest lower bound only applies to sets of real numbers. It does not have a meaningful interpretation in other number systems such as complex numbers or rational numbers.

5. What are some real-world applications of the least upper bound/greatest lower bound?

The least upper bound/greatest lower bound is used in many areas of mathematics and science, such as optimization problems, finding limits of sequences, and proving the convergence of series. It is also used in economics, physics, and engineering to model and solve real-world problems.

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