Finding a relation between two sets

Question

I am using the textbook Elementary Analysis. I am having trouble solving the following problem: Let $A$ be a set of real numbers and let $B=\{-x \mid x \in A\}$. Find a relation between $\max(A)$ and $\min(B)$ and between $\max(B)$ and $\min(A)$. I know that the min of $A$ is the max of $B$ and that the max of $A$ is the min of $B$. I am not sure what it means when it says "find a relation". Any help will be appreciated.

Answer 1

A little bit of graph is always a good thing to do. As you can see in the graphic example, the desired relationship is that $\max A=-\min B$ (we assume that set $A$ has a maximum, although it should have been stated by the assumptions). Indeed, if $a\in A$ is such that $x\leq a$ for all $a\in A$ (i.e. $a=\max A$), then $-x\geq -a$ for all $x\in A$, equivalently $y\geq -a$ for all $y\in B$, meaning that $\min B=-a=-\max A,$ as desired.