MHB An increasing function on the power set of a set

AI Thread Summary
The discussion centers on a function f defined on the power set of a finite set A, which maintains the property that if X is a subset of Y, then f(X) is a subset of f(Y). Participants explore the existence of a set T in the power set such that f(T) equals T. This property is linked to the Knaster–Tarski theorem, which asserts that such a fixed point T exists. The theorem's applicability extends beyond finite sets, making it a significant concept in set theory. Understanding this theorem enhances comprehension of increasing functions on power sets.
caffeinemachine
Gold Member
MHB
Messages
799
Reaction score
15
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.
 
Mathematics news on Phys.org
Evgeny.Makarov said:
This is a special case of Knaster–Tarski theorem.
I didn't know about this theorem. Thanks.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top