Proof of the Equality of Supremums (Or Something Like That Anyway :) )

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In summary, the "Proof of the Equality of Supremums" is a mathematical concept that states that if two sets have the same supremum, they are considered equal. This proof is important in mathematics because it allows for comparison and equating of different sets based on their supremums. It relies on the definition of supremum as the smallest element in a set that is greater than or equal to all other elements. The proof can be applied to any set with a supremum, but it has limitations such as not being able to compare sets without a supremum and only working for sets with a finite number of elements.
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AutGuy98
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Hey guys,

I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says:

"Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so, give an example. If not, prove why not."

Honestly, I'm not even sure where to begin with proving this, so any help would be greatly appreciated on my behalf. Thank you in advance to anyone that replies.
 
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  • #2
AutGuy98 said:
Hey guys,

I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says:

"Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so, give an example. If not, prove why not."

Honestly, I'm not even sure where to begin with proving this, so any help would be greatly appreciated on my behalf. Thank you in advance to anyone that replies.
Hi AutGuy98,

Here is a simple example: take $A=\{1\}$ and $B=\{0,1\}$. You have $\sup A = \sup B = 1$.
 
  • #3


Hi there,

I can definitely understand your struggle with this problem. It can be tricky to wrap your head around at first, but I'll do my best to explain it to you.

To answer the question, yes, it is possible for A to be a subset of B and for A to not equal B, yet have the supremum of A equal the supremum of B. Here's an example:

Let A = {1, 2, 3} and B = {1, 2, 3, 4}

In this case, A is definitely a subset of B because all the elements in A are also in B. However, A does not equal B because B has an additional element, 4, that is not in A. But if we look at the supremum of both A and B, we can see that they are both equal to 3, since that is the highest value in both sets.

Now, to prove why this is the case, we need to understand what the supremum of a set is. The supremum, or least upper bound, of a set is the smallest number that is greater than or equal to all the numbers in the set. In our example, 3 is the smallest number that is greater than or equal to all the numbers in both A and B.

So, even though A and B are not equal sets, they can still have the same supremum because the supremum is not determined by the exact elements in the set, but rather by the upper bound of those elements.

I hope this helps you understand the concept better. Let me know if you have any other questions or need further clarification. Good luck with your problem!
 

Related to Proof of the Equality of Supremums (Or Something Like That Anyway :) )

1. What is the proof of the equality of supremums?

The proof of the equality of supremums is a mathematical proof that shows that the supremum (or least upper bound) of a set of numbers is equal to the supremum of a subset of that set.

2. Why is the proof of the equality of supremums important?

This proof is important because it is a fundamental concept in real analysis and is used to prove many other theorems and properties. It also helps to establish the foundations of calculus and other areas of mathematics.

3. How does the proof of the equality of supremums work?

The proof involves using the definition of supremum and properties of real numbers to show that the supremum of the original set is equal to the supremum of the subset. This is typically done using logical arguments and mathematical equations.

4. Can you provide an example of the proof of the equality of supremums?

Yes, for example, if we have a set A = {1, 2, 3, 4} and a subset B = {2, 3}, the supremum of A is 4 and the supremum of B is 3. The proof would show that 4 is also the supremum of B, as it is the smallest number that is greater than or equal to all elements of B.

5. Are there any applications of the proof of the equality of supremums?

Yes, this proof is used in many areas of mathematics, including real analysis, calculus, and optimization. It is also used in various fields of science and engineering to solve problems and make predictions based on mathematical models.

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