Suppose $X \subset \{1,2,3,\ldots,n\}$. Show that the cardinality of $X$ is $0$, $1$ or $n$, if $\forall$$b \in A_n$, $X \cap bX = \emptyset$ or $X = bX$.
It's pretty clear to me how the cardinality can be 0 or 1. But how do I show that if it isn't 0 or 1, then it must be $n$?
Thanks for your help.