In other words, some texts will commonly use the fact that for any two ordinals we have $\alpha<\beta$ iff $\alpha\in\beta$ and $\alpha\le\beta$ iff $\alpha\subseteq\beta$.
For example, they will simply write union instead of $\sup$, intersection instead of $\inf$.
There are some places where this makes notation much more shorter.
But it can cause confusion for readers who are not familiar with this.
Here is an example of a question which might be related to this:
To show that $\omega_1\leq\frak p$ it suffices to show that $\omega\neq\frak p$. That is, given a countable family with strong finite intersection property, it must have a pseudo-intersection.
So suppose that $\{X_n\in[\omega]^\omega\mid n\in\omega\}$ is a family with the strong finite intersect...
I think that usage of this notation is such prevalent in some areas of mathematics, that you simply need to learn it if you want to read texts from that area.
I'd guess that by browsing a few pages in Halbeisen's book mentioned in that post, we could be able to find many other examples where this notation is used.
Question:
For a surjective function $f：R\to R$,where $R$ be real numbers, define the set
$$S(f)=\{f(x+y)-f(x)-f(y)\ |\ x,y\in R\}\ .$$
Assume that $S(f)$ is finite and $|S(f)|\neq 1$. Find the possible values of $|S(f)|$.
I think $|S(f)|=2$ is possible because of the example
$$f(x)=\d...
Question:
For a surjective function $f：R\to R$,where $R$ be real numbers, define the set
$$S(f)=\{f(x+y)-f(x)-f(y)\ |\ x,y\in R\}\ .$$
Assume that $S(f)$ is finite and $|S(f)|\neq 1$. Find the possible values of $|S(f)|$.
I think $|S(f)|=2$ is possible because of the example
$$f(x)=\d...
