1. ## set theory questions

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.

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?

2. Originally Posted by 2complex4me
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)
Say $\displaystyle A\cap B = \emptyset$. Let $\displaystyle x\in B\subseteq U$ then $\displaystyle x\not \in A$. Thus by definition $\displaystyle x\in U\setminus A$.
Say $\displaystyle B\subseteq U\setminus A$. If $\displaystyle x\in B\subseteq U$ then $\displaystyle x\in U \mbox{ and }x\not \in A$, this mean $\displaystyle A\cap B = \emptyset$.

Q2
Question is a proof by induction... Prove that for any n in w, we have n contained in w.
Note that $\displaystyle \emptyset \subset \omega$. Say that if $\displaystyle n\in \omega \implies n\subset \omega$. Then $\displaystyle n+1 = n \cup \{ n\}$. We note that $\displaystyle n\subset \omega$ and $\displaystyle \{ n\} \subset \omega$ thus $\displaystyle n\cup \{ n\} \subset \omega$. By induction this completes the proof.

3. Q3
Consider any cardinal numbers card(A)=a and card(B)=b and card(C)=c
Prove that (a^b)^c = a^(b*c)
You want to prove: $\displaystyle | (A^B)^C| = |A^{B\times C}|$.
Define a function $\displaystyle \phi : (A^B)^C \mapsto A^{B\times C}$ as follows: let $\displaystyle x \in (A^B)^C$ and define $\displaystyle \phi(x)(c) = x_b(a)$.

4. Originally Posted by ThePerfectHacker
$\displaystyle \phi(x)(c) = x_b(a)$.
Thanks Hacker, but I think i need a bit more of hand here. I'm not sure I understand this notation... Can you please help me a bit further?