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an increasing function on the power set of a set


Well-known member
MHB Math Scholar
Mar 10, 2012
Let $A$ be a finite set. Let $\mathcal{P}(A)$ denote the power set of $A$. Let $f:\mathcal{P}(A)\to \mathcal{P}(A)$ be a function such that $X\subseteq Y\Rightarrow f(X)\subseteq f(Y)$. Show that $\exists T\in \mathcal{P}(A)$ such that $f(T)=T$.

P.S. The power set of $A$ is the set of all the subsets of $A$.
NOTE: The theorem holds even when $A$ is not finite.


Well-known member
MHB Math Scholar
Jan 30, 2012