Are the greatest poet among the painters and the greatest painter among the poets the same person? I have come across this question (based on set theory) in my textbook and I've tried solving it twice but alas, could anybody help me out with it?
Are the greatest painter among the poets and the greatest poet among the painters the same person?
And, quite a similar one,
Are the oldest painter among the poets and the oldest poet among the painters the same person?
For the first one, I suppose that there is a chance that they can't be the same person but failed to come up with a concrete proof with or without a Venn diagram.
Could anybody help me out with it?
For reference, the question is from Sydsaeter, Hammond's Mathematics for Economic Analysis tenth edition
 A: Regarding the first. Perhaps some painters aren't poets and some poets aren't painters. We simply don't know.
Let $A$ be the set of all painters and poets, that is the intersection of the two sets of poets and painters.
We will construct $A$ in such a manner: $A$ consists of pairs of the form $(x,y)$ where $x$ denotes the person's skill in painting and $y$ in poetry. We rated them all from a theoretical scale of $0$ to $100$ (meaning not all values are necessarily occuring), thus $A=\{(x,y)| x,y\in (0,1,\dots,100)\}$.
Let's say that the greatest score on painting skills was $98$ , thus $x=98$. Similarly let's assume that the greatest score on poetry skills was $91$ , thus $y=91$.
There is no reason to assume that these values refer to the same person. That is, we might have $a_1=(98, 73)$ be the greatest painter amongst poets and $a_2=(43, 91)$ the greatest poet amongst painters with $a_1\neq a_2$.
So we have constructed an example where the greatest painter amongst poets and the greatest poet amongst painters are different persons.
It's easy to see that this condition holds for our example: $\forall a_i \in A, 0\le x\le98 $ and $0\le y\le91$
Then again, it might very well be that if $\max x = \max y$ $\forall a_i \in A$, then indeed the greatest painter among the poets and the greatest poet among the painters is the same person but it is not necessarily so.
