Understanding Sup&Inf: "For any two sets of real numbers

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In summary: X and an element of Y. For example, if X=[0,2] and Y=[1,3], then x-y=[-3,2]. This is different from the set difference, which is only the elements of X that do not appear in Y. The notations for supremum and infimum can also be written as \sup A and \inf A, where A is the set. In summary, the quote from the textbook is discussing the properties of the supremum and infimum of sets of real numbers, and clarifies the notation used in the equations (1) and (2
  • #1
Kamataat
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Quote from my textbook:

"For any two sets of real numbers, [itex]X=\{x\}[/itex] and [itex]Y=\{y\}[/itex], the following hold:

[tex]\sup(x-y)=\sup x-\inf y[/tex] (1),
[tex]\inf(x-y)=\inf x-\sup y[/tex] (2),

where [itex]x\in X[/itex] and [itex]y\in Y[/itex]."

It is also said, that if some set [itex]A[/itex] consists of some elements [itex]a[/itex], we may write it as [itex]A=\{a\}[/itex], and [itex]\sup a[/itex] is another way of writing [itex]\sup A[/itex].

Now the above quote is confusing. If I take [itex]X=\mathbb{R}[/itex] and [itex]Y=[10;+\infty)[/itex], then [itex]X-Y=(-\infty;10)[/itex]. Then the supremum of the difference of the sets is 10, but the infimum subtracted from the supremum is not 10, like it should be according to (1).

Can anybody clear that quote up for me?

PS: It is assumed in this textbook that the supremum (infimum) of the set of all real numbers is [itex]+\infty[/itex] ([itex]-\infty[/itex]).

- Kamataat
 
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  • #2
x-y in (1) and (2)
does not mean X-Y the set difference
the largest subset of X which has an empty intersection with Y
x-y mean the set of all numbers that can be written element of x-element of y
So if X=[0,2] Y=[-1,1]
X-Y=(0,2]
x-y=[-1,3]
A few confusions result when a symbol such as "-" is used to mean so many different things.
 
Last edited:
  • #3
Yeah, the notation was confusing me. BTW, I think you mean X-Y=(1,2].

Thanks!

- Kamataat
 

1. What is the definition of supremum and infimum?

The supremum of a set of real numbers is the smallest upper bound, while the infimum is the largest lower bound. In other words, the supremum is the least number that is greater than or equal to all elements in the set, and the infimum is the greatest number that is less than or equal to all elements in the set.

2. How do you determine the supremum and infimum of a set of real numbers?

To determine the supremum and infimum of a set of real numbers, you can use the following steps:

  • Arrange the numbers in the set in ascending order
  • The supremum is the last number in the set
  • The infimum is the first number in the set

3. Can a set have more than one supremum or infimum?

Yes, a set can have multiple supremum and infimum if there are multiple numbers that satisfy the definition. For example, in the set {1, 2, 3}, both 2 and 3 are supremums, and both 1 and 2 are infimums.

4. What is the significance of supremum and infimum in real analysis?

Supremum and infimum are important concepts in real analysis because they help us understand the bounds and limits of a set of numbers. They also play a crucial role in defining important concepts such as limits, continuity, and convergence of sequences and series.

5. How do supremum and infimum relate to maximum and minimum?

The supremum is the smallest upper bound, while the maximum is the greatest element in a set. Similarly, the infimum is the largest lower bound, while the minimum is the smallest element in a set. In some cases, the supremum or infimum can also be the maximum or minimum, but this is not always the case.

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