Is there anything wrong with how the supremum of a set is written?

In summary, the conversation discusses the syntax of writing the supremum of a set and whether it exists for a continuous function and a compact set. The participants also mention the requirement of the function being bounded above in order for the supremum to exist. An example is given where a function on a half-open interval can have infinite values at the endpoint, but this cannot occur if the domain is compact.
  • #1
Eclair_de_XII
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TL;DR Summary
Let ##A\subsetℝ## be bounded from above. Let ##f:A\rightarrow ℝ##. Then can I write the supremum of the image of ##A## through ##f## as ##\sup f(A)##, or do I have to write it as ##\sup\{f(x):x\in A\}##?
I'm just having random thoughts today, and I didn't know where to put this, since this isn't even a homework problem.

Anyway, is my way of writing the supremum of a set correct syntax-wise, or no?
 
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  • #2
Both are the same thing: simply because ##f(A) = \{f(a)\mid a \in A\}## by definition. That A is bounded above does not imply that f(A) is bounded above, so the supremum may not exist as a real number.
 
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  • #3
I think having a continuous ##f## and compact ##A## would guarantee the existence of a supremum (probably even maximum).
 
  • #4
hilbert2 said:
I think having a continuous ##f## and compact ##A## would guarantee the existence of a supremum (probably even maximum).

Yes, that's work. You are right that it is even a maximum because the image of a compact set under a continuous map is compact, so the image is in particular closed and bounded hence contains its supremum.

What we need here is the requirement that f is bounded above, but this is just a reformulation of f(A) is bounded above.
 
  • #5
Math_QED said:
That A is bounded above does not imply that f(A) is bounded above

Oh, I should have caught that. My mista;e; thanks for the reply, besides.
 
  • #6
A situation where that happens is the function (defined on a half-open interval)

##f:]0,1]\mapsto \mathbb{R}## such that ##f(x)=1/x##

which gets arbitrarily large values when approaching ##x=0##.

If the domain is compact (closed and bounded), this can't happen because in conventional calculus you can't have a function that is defined as "infinite" at an endpoint of the domain.
 

1. What is the supremum of a set?

The supremum of a set is the least upper bound, or the smallest number that is greater than or equal to all the elements in the set.

2. How is the supremum of a set written?

The supremum of a set is typically written as sup(A) or sup{A}, where A is the set in question.

3. Is there a specific way to write the supremum of a set?

There is no specific or standardized way to write the supremum of a set. Some mathematicians use sup(A), while others use sup{A} or simply write "the supremum of A". It is important to be consistent within a particular context or mathematical proof.

4. Can the supremum of a set be written as a decimal or fraction?

Yes, the supremum of a set can be written as a decimal or fraction if the elements of the set are real numbers. However, if the set contains complex numbers, the supremum may be written in terms of the imaginary unit, i.e. sup(A) = a + bi, where a and b are real numbers and i is the imaginary unit.

5. Is there a difference between the supremum of a set and the maximum value of a set?

Yes, there is a difference between the supremum of a set and the maximum value of a set. The supremum is the smallest number that is greater than or equal to all the elements in the set, while the maximum value is the largest number within the set. The supremum may or may not be an actual element of the set, whereas the maximum value must be an element of the set.

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