Correcting a Previous Blog Post
by djmathman, Dec 8, 2018, 3:45 AM
On January 4th of this year, I posted a solution to the following problem from my Combo 3 notes back at AwesomeMath 2013.
Problem. Let
be a nonempty set, and let 
be an increasing function on the set of subsets of , meaning that 
if 
. Prove that there exists 
, a subset of , such that 
.
Unfortunately, it turns out that my solution is dead wrong. In the proof that my chosen set satisfied
, I said
This sentence is quite bogus, as 
is just the value of the function 
at , not the image ![$f[T] = \{f(x):x\in T\}$](//latex.artofproblemsolving.com/d/4/4/d44712db41e54c2dbed5a2e0e1f2e1fe381c3228.png)
(which doesn't even typecheck!). The rest of the solution is rendered invalid as a result, and I actually believe the approach outlined in the solution is doomed to fail.
Here is, as far as I can tell, an actual solution.
Problem. Let
be a nonempty set, and let 
be an increasing function on the set of subsets of , meaning that 
if 
. Prove that there exists 
, a subset of , such that 
.Unfortunately, it turns out that my solution is dead wrong. In the proof that my chosen set satisfied
, I said
Quote:
Let 
, so that there exists 
such that 
.
, so that there exists 
such that 
.
is just the value of the function 
at , not the image ![$f[T] = \{f(x):x\in T\}$](http://latex.artofproblemsolving.com/d/4/4/d44712db41e54c2dbed5a2e0e1f2e1fe381c3228.png)
(which doesn't even typecheck!). The rest of the solution is rendered invalid as a result, and I actually believe the approach outlined in the solution is doomed to fail.Here is, as far as I can tell, an actual solution.
the collection of subsets 
such that 
;
.
,
.
,
of all elements in 
.
has an upper bound in 
is some chain in ![\[
M = \bigcup_{C\in\mathcal C} C.
\]](http://latex.artofproblemsolving.com/e/b/0/eb0a10ba76c8a61b257a4c6ec5c0b55de8356d37.png)
Note that 
for all 
,
is certainly larger (in the sense of the partial order) than all the other elements in 
.
;
yields![\[
f(M)\supseteq \bigcup_{C\in\mathcal C} C= M,
\]](http://latex.artofproblemsolving.com/8/4/8/848d5b50fe897ebb3ebfd9fa529c9abe9b2df78a.png)
and so 
,
,This post has been edited 5 times. Last edited by djmathman, Dec 8, 2018, 3:49 AM
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