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Math Help - set theory questions

  1. #1
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    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.

    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?
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  2. #2
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    Quote Originally Posted by 2complex4me View Post
    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 A\cap B = \emptyset. Let x\in B\subseteq U then x\not \in A. Thus by definition x\in U\setminus A.
    Say B\subseteq U\setminus A. If x\in B\subseteq U then x\in U \mbox{ and }x\not \in A, this mean 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 \emptyset \subset \omega. Say that if n\in \omega \implies n\subset \omega. Then n+1 = n \cup \{ n\}. We note that n\subset \omega and \{ n\} \subset \omega thus n\cup \{ n\} \subset \omega. By induction this completes the proof.
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  3. #3
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    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: | (A^B)^C| =  |A^{B\times C}|.
    Define a function \phi : (A^B)^C \mapsto A^{B\times C} as follows: let x \in (A^B)^C and define \phi(x)(c) = x_b(a).
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    \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?
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