Hello,
I hope my set theory questions are in the right section of your forum..
I have just finished my exams and now have time to take a look at a couple of problems that caught me out during semester. These are probably really elementary questions, but proving things has never been a strong point, so here goes...
Questions 1,2,3 are most important to me as I have a faint hope that my answers to 4,5,6 are ok.
Thank you for your clues...
Q1
Note that A,B,U are sets such that A,B are contained in U
A "intersection" B = {} <=> B "is contained in" (U\A)
I see that this is very obviously true, but apart from assuming LHS for for proving =>, and assuming RHS for proving <=, I don't really know what to do, any clues would be great.
Q2
Consider the Von Neumann definition of the natural numbers.
w = {0,1,2,3,...} is the set of such numbers
w is an inductive set => 0={} is in w and for all k in w, k+ is in w
k = {0,1,...,k-1}
k+ = k union {k}
Sorry for the preamble there, just wanted to make sure we're all on the same page.
Question is a proof by induction... Prove that for any n in w, we have n contained in w.
The base step is done, I'm stuck at the inductive step.
All I can think of so far is what you do at the start of every inductive step...
Assume that for some n=k in w, we have n=k is contained in w
RTP n=k+ is contained in w
Clues would be great.
Q3
Consider any cardinal numbers card(A)=a and card(B)=b and card(C)=c
Prove that (a^b)^c = a^(b*c)
I understand that to do this, I need to prove an equivalence of sets by finding a bijection
f: (A^B)^C --> A^(BxC)
I dont really have a clue as to what that f should be...
Am I correct in saying:
(A^B)^C is the set of all maps from C to the set of all maps from B to A, and
A^(BxC) is the set of all maps from B cross C to A
??
Q4
Give a partially ordered set (X,<=) such that X has two elements and (X,<=) has no greatest (maximum) element
I chose <= to be the standard set containment relation and
X = { {x},{y} } for any x ~= y
Is this ok?
Q5
Give a partially ordered set with a minimal element but no least element
I chose
(X,<=) where
X = { {x} , {y} , {x,y} } for any x ~= y
and <= is the standard set containment relation
So clearly {x} and {y} are minimal, {x,y} is both maximal and maximum and there is no minimum ???
Is this ok?
Q6
A partially ordered set with no maximal element
I chose (X,<=)
Where X = [0,1) and <= is the standard <= on the reals.
Is this ok?