Sup&inf question

  • Thread starter Kamataat
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  • #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
 

Answers and Replies

  • #2
lurflurf
Homework Helper
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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
Kamataat
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Yeah, the notation was confusing me. BTW, I think you mean X-Y=(1,2].

Thanks!

- Kamataat
 

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