Given a set $A$ such that for any family of sets $F$: if $\cup F = A$ then $A \in F$.
Prove that $A$ has only one element.
Please help.
Thank you all.
I tried to solve it by assuming that there are two elements in $A$. But I can't solve it.
Given a set $A$ such that for any family of sets $F$: if $\cup F = A$ then $A \in F$.
Prove that $A$ has only one element.
Please help.
Thank you all.
I tried to solve it by assuming that there are two elements in $A$. But I can't solve it.
Well: if $\,|A|\geq 2\,$ then the family $\,\mathcal F:=\{\,\{a\}\;\;;\;\;a\in A\}\,$ fulfills the given condition
$\bigcup_{F\in\mathcal F}F=A\,\,\,,\,\,\text{yet}\,\,A\notin\mathcal F\,\;\;(\text{can you see why?)}$
Thus, it must be that $\,|A|=1\,$