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Math Help - Topology beginer some help with proofs please....

  1. #1
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    Topology beginer some help with proofs please....

    Hi,

    I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs.

    The text in well presented but to get a proper understanding I am working through the excercises. I am still on Chapter 1 which deals with sets and for the most part the excercises seem trivial to the point where its seems to me no proof is needed BUT there are a couple of questions that have me stumped (there are no worked solutions a problem with most maths books I find).

    I would really apreciate it if somebody could walk me through a few proofs (it will probably take more than a couple before I get my head into gear), so here a couple of the excercises..

    First one that seems trivial to me but one I am having trouble forming a proof for,

    Let A, B, D \subset S

    Prove A \cap (B \cup D) = (A \cap B) \cup (A \cap D)

    Secondly not so trivial to my mind,

    Let \{A_\alpha\}_{\alpha\in I} be an indexed family of subsets of a set S. Let J \subset I

    Prove \bigcap_{\alpha\in J} A_\alpha \supset \bigcap_{\alpha\in I} A_\alpha

    Any help with this most apreciated.
    Last edited by Bwts; July 24th 2012 at 05:26 AM.
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  2. #2
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    Re: Topology beginer some help with proofs please....

    The "standard" way to prove set X= set Y is to show "X is a subset of Y" and "Y is a subset of X". And the "standard" way to show "X is a subset of Y" is to start "if x is in set X" and use the definitions of X and Y to conclude "x is in Y".

    So: If x\in A\cap(B\cup D) then x\in A and [tex]x\in B\cup D/tex]. Since x\in B\cup D, either x\in B or x\in D.
    case 1) x\in B. Since x\in A and x\in B, x\in A\cap B.

    case 2) x\in D. Since x\in A and x\in D, x\in A\cap D.

    In either case, x\in (A\cap B)\cup (A\cap D) so A\cap(B\cup D)\subset (A\cap B)\cup (A\cap D).

    Now, if x\in (A\cap B)\cup (A\cap D) then either x\in A\cap B or x\in A\cap D.
    case 1) x \in A\cap B. Then x\in A and x\in B. Since x\in B, x\in B\cup D. Since [tex]x\in A[tex] and x\in B\cup D, x\in A\cap(B\cup D).

    case 2) x\in A\cap D. Then x\in A and x\in D. Since x\in D, x\in B\cup D. Since [tex]x\in A[tex] and x\in B\cup D, x\in A\cap(B\cup D).
    Thanks from pratique21 and Bwts
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    Re: Topology beginer some help with proofs please....

    For the second, \{A_\alpha}_{\alpha\in I}\} and J a subset of I, start the same way:
    If x\in \cap_{\alpha\in J}A_\alpha then x\in A_\beta for \beta\in J. But J is a subset of I so \beta\in I.
    That is, A_\beta\subset \cap_{\alpha\in I}A_\alpha thus x\in \cap_{\alpha\in I}A_\alpha.
    Last edited by HallsofIvy; July 24th 2012 at 08:16 AM.
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    Re: Topology beginer some help with proofs please....

    Quote Originally Posted by HallsofIvy View Post
    For the second, \{A_\alpha}_{\alpha\in I}\} and J a subset of I, start the same way:
    If x\in \cap_{\alpha\in J}A_\alpha then x\in A_\beta for \beta\in J. But J is a subset of I so \beta\in I.
    That is, A_\beta\subset \cap_{\alpha\in I}A_\alpha thus x\in \cap_{\alpha\in I}A_\alpha.
    unfortunately, it looks like you proved the wrong thing here.

    suppose that: x \in \bigcap_{\alpha \in I} \{A_\alpha\}.

    this means that x \in A_\alpha for every \alpha \in I.

    since J \subset I, this means that for every \beta \in J,\ A_\beta = A_\alpha for some \alpha \in I.

    since x \in A_\alpha for every \alpha \in I, certainly x \in A_\beta for every \beta \in J.

    thus: x \in \bigcap_{\beta \in J} \{A_\beta\}.
    Thanks from Bwts
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    Re: Topology beginer some help with proofs please....

    Thankyou both. I will work through some more tomorrow and see if I can get into thinking in this way.

    Something that struck me when reading HallsofIvy's first post was a similarity to Boolean algebra in places \cup for 'AND' and \cap for 'XOR', complement of a set 'NOT' etc..

    Then I remembered using De Morgan's theorem for reducing logic circuits, so I am guessing there is a big cross over?
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  6. #6
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    Re: Topology beginer some help with proofs please....

    The similarity between laws for sets and for propositions comes from the fact that x\in A\cup B\Leftrightarrow x\in A\lor x\in B and similarly for \cap and \land.
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    Re: Topology beginer some help with proofs please....

    yes. one can make the following correspondence:

    union = or
    intersection = and
    subset = implies
    complement = not
    symmetric difference = xor
    is a member = true

    that is, both logical propositions and statements about sets can be formulated in terms of boolean algebra.

    and what is germane to topology here, is that "subset" (containment) gives a partial order on the lattice of the power set of a topological space X, which carries over to the sub-lattice of open sets of X. so a lot of theorems in topology consist in merely "translating" logic to sets.
    Thanks from Bwts
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    Re: Topology beginer some help with proofs please....

    Thanks I'm making some headway with this now.

    I am struggling with the following though (I seem to be struggling with anything with a dummy index or rather 2 indices defines on the same set?)

    Let \{A_\alpha\}_{\alpha \in I} and \{B_\alpha\}_{\alpha \in I} be two indexed families of subsets of a set S,

    Prove for each \beta \in I, A_\beta \subset \bigcup_{\alpha \in I}A_\aplha

    I know its somethimg obvious but just keeps eluding me.

    Let \{A_\alpha\}_{\alpha \in I} be an indexed family of subsets of a set S and let B \subset S

    Prove

    a) B \subset \bigcap_{\alpha \in I}A_\alpha if and only (iff?) for each \beta \in I, B \subset A_\alpha

    b) \bigcup_{\alpha \in I}A_\alpha \subset B if and only if for each \beta \in I, A_\beta \subset B

    There is one more trickier question that is puzzling me but I think I can work it out if I can see how the above work.

    Thanks again for patientce, I'm finding this whole thing facinating and you are really help me out.
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    Re: Topology beginer some help with proofs please....

    Quote Originally Posted by Bwts View Post
    Let \{A_\alpha\}_{\alpha \in I} and \{B_\alpha\}_{\alpha \in I} be two indexed families of subsets of a set S,

    Prove for each \beta \in I, A_\beta \subset \bigcup_{\alpha \in I}A_\aplha
    Suppose a company of soldiers consists of three platoons. Is is not obvious to you that platoon #2 is a subset of the company?

    Quote Originally Posted by Bwts View Post
    Let \{A_\alpha\}_{\alpha \in I} be an indexed family of subsets of a set S and let B \subset S

    Prove

    a) B \subset \bigcap_{\alpha \in I}A_\alpha if and only (iff?) for each \beta \in I, B \subset A_\alpha
    Have you tried applying the definitions of subset and intersection?
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    Re: Topology beginer some help with proofs please....

    Thanks for the platoon analogy that worked

    Ok this is how far I got with the next one,

    If x \in \bigcap_{\alpha \in I} A_\alpha and x \in A_\beta then A_\beta \subset \bigcap_{\alpha \in I} A_\alpha

    ..now it gets a bit sketchy... how do I introduce B?

    am I on the right track?
    Last edited by Bwts; July 26th 2012 at 06:49 AM.
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  11. #11
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    Re: Topology beginer some help with proofs please....

    Quote Originally Posted by Bwts View Post
    If x \in \bigcap_{\alpha \in I} A_\alpha and x \in A_\beta then A_\beta \subset \bigcap_{\alpha \in I} A_\alpha
    I am assuming you are proving the left-to-right direction:

    If B \subset \bigcap_{\alpha \in I}A_\alpha, then B\subset  A_\beta for each \beta\in I. (*)

    First, it is not clear what \alpha and \beta in your formulas are. Did you choose them arbitrarily or using some principle? Also, why do you assume x\in A_\alpha? How are you going to get rid of this assumption (since the main claim does not have it)? Finally, A_\beta \subset x \in \bigcap_{\alpha \in I} A_\alpha does not make sense. It can't mean A_\beta \subset x because x does not have to be a set (typically, x ranges over points of some topological space and A_\alpha ranges over sets of points). It also can't mean A_\beta \subset \left(x \in \bigcap_{\alpha \in I} A_\alpha\right) because x \in \bigcap_{\alpha \in I} A_\alpha is a proposition, not a set.

    To begin with, you need an intuitive understanding why this claim is true. Suppose I is a set of classes and if \alpha\in I, then A_\alpha denote the set of students taking class \alpha. What does \bigcap_{\alpha\in I}A_\alpha represent? If B is a set of three guys leaving in room 528 and B\subset\bigcap_{\alpha\in I}A_\alpha, what does it mean?

    Next, please write the definitions of \subset and \bigcap in terms of ∈:

    A\subset B if ...
    x\in\bigcap_{\alpha\in I}A_\alpha if ...

    Finally, start proving (*) one step at a time. To prove "P implies Q", you assume P and prove Q. To prove "for all x, P(x)", you fix an arbitrary x and prove P(x). If you have an assumption of the form "for all x, P(x)", you can instantiate x with any well-formed object. So, the first several step in the proof of (*) would be the following.

    (1) Assume B \subset \bigcap_{\alpha \in I}A_\alpha; need to prove B\subset  A_\beta for each \beta\in I.
    (2) Fix an arbitrary \beta\in I; need to prove B\subset  A_\beta.
    (3) Using the definition of \subset, rewrite B\subset  A_\beta as "for all x, ..."

    Take special care not to use any variable that you don't properly introduce. Each variable must be introduced in one of three ways:

    (a) "fix x" as a part of the proof of "for all x, ..."
    (b) "fix x such that P(x)" if you have an assumption of the form "there exists an x such that P(x)"
    (c) "let x be ..."; you have to provide an explicit definition for x.
    Thanks from Bwts
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    Re: Topology beginer some help with proofs please....

    some things that may be useful:

    we are working on 3 "levels" of organization:

    points of a space X, such as x,y,z, etc. there is, in general, very little restriction on what these "points" might actually be. in the most common examples, they are actually "points" like the point (2,5) in the cartesian plane, but they might people as in the set {Thomas, Betty, Ed}, or functions such as the set {L: L is a line through the origin in R3}, or a variety of other entities.

    collections of points, organized into sets. these are often denoted A,B,C, etc. what are going to be the main sets of interest (in topology) are open sets containing a certain point x, called neighborhoods of x. note that the "index sets" also belong on this level, in fact an "indexed set" is nothing more than a function

    f:I→2X

    from the indexing set to the set of subsets of X, in other words f(i) = Ai.

    collections of sets (also called families of sets). often these sets are indexed by an "outside" set. one common index is the natural numbers, in which case {Aα} can be thought of as the more familiar {A1,A2,A3,....}. but we might have "too many A" for the natural numbers to work as an index.

    for example, we might want to consider the family of open disks of radius 1 centered at every point (x,y) in the plane. now, there are uncountably many points in the plane (our index set is R2), so we can't just list these as B1, B2, B3,...etc. we have to tag these as something like B(x,y). in some texts, families of sets are denoted by script letters, or bold-face (the topologies themselves are one example of these kinds of families), to indicate we're on "the third level up".

    the "grand-daddy" of all families of sets in X, is, of course, 2X, the power set of X. it is easy to get confused and think the singleton set {x} is the same as the point x.

    the other mistake that is easy to make is where the space X lies: it is on "the middle level", because it itself is just the collection of all its "points". in other words, a topology on X lies in the level "above X" (families of sets), so it is a kind of "superstructure on X".

    because of this "multi-layering", you need to be careful about distinguishing between "∈" (is an element of) and "⊆" (is a subset of):

    x ∈ X
    {x} ∈ 2X
    {x} ⊆ X

    elements lie "just one layer down", subsets lie "on the same level" (usually the middle level, but not always: for example a topology τ on X is a subset of 2X, in other words, both of these are on "the third level").

    in proving A ⊆ B, one usually goes "one level down", and shows that x∈A leads to x∈B. what you are doing in your exercises (for some of them, at least), is proving relationships between families by passing one level down to relationships between sets. sometimes you have to do this *twice* to get to the individual points.
    Thanks from Bwts
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    Re: Topology beginer some help with proofs please....

    Quote Originally Posted by emakarov View Post
    To begin with, you need an intuitive understanding why this claim is true. Suppose I is a set of classes and if \alpha\in I, then A_\alpha denote the set of students taking class \alpha. What does \bigcap_{\alpha\in I}A_\alpha represent?
    The set of students who are taking every class?

    If B is a set of three guys leaving in room 528 and B\subset\bigcap_{\alpha\in I}A_\alpha, what does it mean?
    B is a subset of the students who are taking all the classes?

    Next, please write the definitions of \subset and \bigcap in terms of ∈:

    A\subset B if ...
    A \subset B if for all x \in A then x \in B is this right?

    x\in\bigcap_{\alpha\in I}A_\alpha if ...
    x\in\bigcap_{\alpha\in I}A_\alpha if for any \beta \in I, x \in A_\beta ?
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  14. #14
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    Re: Topology beginer some help with proofs please....

    Oll Korrect so far.
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    Re: Topology beginer some help with proofs please....

    Prove B \subset \bigcap_{\alpha \in I} A_\alpha if and only if for each \beta \in I, B \subset A_\alpha.

    If B \subset \bigcap_{\alpha \in I} then for every x \in B, x \in \bigcap_{\alpha \in I} A_\alpha

    If x \in \bigcap_{\alpha \in I} A_\alpha then for every \beta \in I, x \in A_\beta

    So x \in B if and only if B \subset A_\beta

    For that last line can i just write..

    So x \in B if and only if B \subset A_\alpha as \beta is a dummy index?
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