For a finite set $X$ let $X^* $ be the set of all non-empty proper subsets of $X$. Let $f : X^* \longrightarrow X^* $ be an increasing function such that for some $A \in X^* $, $|f(A)| \not = |A|$. It is true that $f$ must have a fixed point ?

(By increasing I mean when $A\subseteq B$ then $f(A)\subseteq f(B)$ )