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## Main Question or Discussion Point

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

"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