Real Analysis: Struggling with 6 Questions? Get Expert Help Now!

[katyat]

I am stuck on 6 question and I can't figure them out. I have an exam coming up and were given 20 questions for review that we should know how to do. However, I am stuck on 6. Please help me.

 

[Bacle]

Hi, katyat:

AFAIK, given sets S,T, to show |S|<|T|, you just need to show there is an injection
from S into T. If what you need is the more strict |S|<|T|, and |S|=/ |T|, you need
to show there is no injection between T and S. Can you see some injections that seem
immediate, and why T cannot inject into S.?

 

[katyat]

no i can't see it

 

[katyat]

how about for the other questions...any suggestions?

 

[Bacle]

Well, think of "embedding" an element as a subset. Use a small set like S={1,2,3} , construct P(S) and I think you will see how you can "embed" S in P(S). Then you
can extend this to any set, including infinite sets.

For the set of points of continuity, the argument I know uses the oscillation of a
function over an interval : Osc(f,p) is the oscillation of f at the point p, defined by:

Osc(f,p)=sup|f(x)-f(y)| over all points in neighborhoods containing p. Then f is
continuous at p if Osc(f,p)<1/n for every natural n. Try using this to construct/define
an infinite sequence of open sets (indexed by n , as in 1/n), such that p must
belong to each of them (which means p belongs to their intersection...) for f
to be continuous at p

Note that this oscillation is just a reformulation of the delta-epsilon def. of continuity.


For #3, I have not tried this yet, but try using the definition of Caratheodory measurability and the def. of symmetric difference of sets (which is used in the
first problem.).


For the Cantor set , notice that at each step you are removing an open interval
you can tell what the measure of each removed interval is. You end up with a disjoint
collection of open intervals
Also:what can you say (re basic topology)about the complement of an open interval..
Maybe expressing terms in Cantor set in base 3 will help show that the complement is
dense in the Cantor set C. Take a point p in the complement and consider any ball about
p, show it must contain points of C.

Sorry, got to go. I will be back tomorrow or thursday.

 

[JSuarez]

Question 6 is Cantor's Theorem: you must prove that, if f:X\rightarrow P(X) is injective, then it cannot be surjective. This is proved by diagonalization: consider the set A \in P(x), defined by:

A=\left\{x \in X:x\notin f(x)\right\}

Now, prove that there is no set B \subseteq X, such that f(B) = A.