Suppose we have a countable set $X$, say $X=\mathbb{N}$, and let $Q \colon X \rightarrow \{0,1\}$ be a function. Is $$\min \{x \in X \colon Q(x)=1\}$$ the same as $$ \inf \{x \in X \colon Q(x)=1\}.$$
What about $\max$ and $\sup$? And for uncountable sets it is not the same?
Thanks for your help.
On
It depends on the ordering of $X$. If $X$ is well ordered, I.E. Every subset has a least element, then inf and min agree in general. The naturals are a canonical example of a well ordering. But if $X$ is something like the integers or reals, then the minimum or inf might not always be defined. In the real case, for example, if $Q$ is the characteristic function of $(0,1)$, then you have an inf but no min.
The above discussion works with sup and max as well.
This has less to do with countable vs. uncountable as with the question whether $X$ well-ordered. In a well-ordered set, every non-empty subset has a minimal element. On the other hand for example $X:=\{\,\frac1n:n\in\mathbb N\,\}$ does not have a minimal element (nor does any infinite subset of $X$) even though it is countable.